I 


CIHM 

Microfiche 

Series 

(i\/lonographs) 


ICIVIH 

Collection  de 
microfiches 
(monographies) 


0 


Canadian  Institute  for  Historical  MIcroreproductlon.  /  Institut  Canadian  de  microreproductions  historique. 


0 


lOx 


sfp*?*- '  -^TTr^'BrsT-" 


^^M^^^^^' '' '  ''^^■^-^j'   -^^-^iiifc  ■ '^''ifey^^-fe.^Si^il 


Technical  and  Bibliographic  Notes  /  Notes  techniques  et  bibliographiques 


The  Institute  has  attempted  to  obtain  the  best  original 
copy  available  for  filming.  Features  of  this  copy  which 
may  be  bibliographically  unique,  which  may  alter  any  of 
the  images  in  the  reproduction,  or  which  may 
significantly  change  the  usual  method  of  filming  are 
checked  below. 


n 
n 

a 


n 


Coloured  covers  / 
Couverture  de  couleur 

Covers  damaged  / 
Couverture  endommag6e 

Covers  restored  and/or  laminated  / 
Couverture  restaurde  et/ou  peliicul^e 

Cover  title  missing  /  Le  titre  de  couverture  manque 

Coloured  maps  /  Cartes  g6ographiques  en  couleur 

Coloured  ink  (i.e.  other  than  blue  or  black)  / 
Encre  de  couleur  (i.e.  autre  que  bleue  ou  noire) 

Coloured  plates  and/or  illustrations  / 
Planches  et/ou  illustrations  en  couleur 

Bound  with  other  material  / 
Relie  avec  d'autres  documents 

Only  edition  available  / 
Seule  Edition  disponible 

Tight  binding  may  cause  shadows  or  distortion  along 
interior  margin  /  La  reliure  serree  peut  causer  de 
I'ombre  ou  de  la  distorsion  le  long  de  la  marge 
int^rieure. 

Blank  leaves  added  during  restorations  may  appear 
within  the  text.  Whenever  possible,  these  have  been 
omitted  from  filming  /  II  se  peut  que  certaines  pages 
blanches  ajout6es  lors  d'une  restauration 
apparaissent  dans  le  texte,  mais,  lorsque  cela  etait 
possible,  ces  pages  n'ont  pas  et6  film§es. 

Additional  comments  / 
Commentaires  suppl^mentaires: 


L'Institut  a  microfilm^  le  meilleur  exemplaire  qu'il  lui  a 
^t^  possible  de  se  procurer.  Les  details  de  cet  exem- 
plaire qui  son!  peut-6tre  uniques  du  point  de  vue  bibli- 
ographique,  qui  peuvent  modifier  une  image  reproduite, 
ou  qui  peuvent  exiger  une  modification  dans  la  m^tho- 
de  normale  de  filmage  sont  indiqu^s  ci-dessous. 

I     I  Coloured  pages  /  Pages  de  couleur 

I I   Pages  damaged  /  Pages  endommag6es 


n 


Pages  restored  and/or  laminated  / 
Pages  restaur^es  et/ou  pellicul^es 


r~^   Pages  discoloured,  stained  or  foxed  / 
I  V)   Pages  d^color^es,  tachet^es  ou  piqu^es 

Pages  detached  /  Pages  d6tach6es 

/     Showthrough  /  Transparence 

I      I   Quality  of  print  varies  / 


n 

D 


n 


Quality  in^gale  de  I'impression 

Includes  supplementary  material  / 
Comprend  du  materiel  supplement?  e 

Pages  wholly  or  partially  obscured  by  i  ia  sIVs, 
tissues,  etc.,  have  been  refilmed  to  ensji :  ■  h  jst 
possible  image  /  Les  pages  totale.;.-nt  ou 
partiellement  obscurcies  par  un  feuillet  d'errata,  une 
pelure,  etc.,  ont  6t6  film6es  k  nouveau  de  fafon  k 
obtenir  la  meilleure  image  possible. 

Opposing  pages  with  varying  colouration  or 
discolourations  are  filmed  twice  to  ensure  the  best 
possible  image  /  Les  pages  s'opposant  ayant  des 
colorations  variables  ou  des  decolorations  sont 
film^es  deux  fois  afin  d'obtenir  la  meilleure  image 
possible. 


This  item  is  filmed  at  the  reduction  ratio  checked  below  / 

Ce  document  est  (ilme  au  taux  de  reduction  indlqu6  ci-dessous. 


lOx 

14x 

18x 

22x 

26x 

30x 

V 

12x 


16x 


20x 


24x 


28x 


32x 


Th*  copy  filmed  h«r«  has  bMn  raproducad  thanks 
to  tha  ganareaity  of: 

National  Library  of  Canada 


L'axampiaira  filmi  fut  raproduit  grica  k  la 
gintresiti  da: 

Bibliotheque  nationale  du  Canada 


Tha  imagas  appaaring  hara  ara  tha  bast  i|uallty 
possibia  considaring  tha  condition  and  lagibility 
of  tha  original  copy  and  in  kaaping  with  tha 
filming  contract  spacif ications. 


Las  imagas  suivantas  ont  *ti  raproduitas  avac  la 
plus  grand  soin,  compta  tanu  da  Is  condition  at 
da  la  nanati  da  l'axampiaira  filmi,  at  tn 
conformity  avac  laa  conditions  du  contrat  da 
filmaga. 


Original  copias  in  printad  papar  covars  ara  fllmad 
baginning  with  tha  front  covar  and  anding  on 
tha  last  paga  with  a  printad  or  illustratad  impraa* 
sion.  or  tha  back  covar  whan  approprlata.  All 
othar  original  copias  »rm  filmad  baginning  on  tha 
first  paga  with  a  printad  or  illustratad  impras- 
sion,  and  anding  on  tha  last  paga  with  a  printad 
or  illuatratad  imprassion. 


Laa  axamplairas  originaux  dont  la  couvarturs  sn 
papiar  ast  imprimaa  sont  filmSs  sn  commsn^ant 
par  la  pramiar  plat  at  mn  tarminant  soit  par  la 
darniara  paga  qui  comporta  una  amprainta 
d'imprassion  ou  d'illustration,  soit  par  la  tacond 
plat,  salon  la  cas.  Toua  laa  autras  axamplairas 
originaux  sont  filmOs  mn  commandant  par  la 
pramiOra  paga  qui  comporta  una  amprainta 
d'imprassion  ou  d'illustration  at  an  tarminant  par 
la  darniira  paga  qui  comporta  una  talia 
amprainta. 


Tha  last  racordad  frama  on  aach  microficha 
shall  contain  tha  symbol  «-<^  (moaning  "CON- 
TINUEO").  or  tha  symbol  V  (moaning  "END"), 
whichavar  appiias. 


Un  daa  symbolos  suivants  spparaitrn  sur  la 
darniira  imaga  da  ehaqua  microficha.  salon  la 
cas:  la  symbols  — »  signifia  "A  SUIVRE  ".  la 
symbolo  Y  signifia  "FIN  ". 


Maps,  platas,  chans,  ate,  may  ba  filmad  at 
diffarant  reduction  ratios.  Thosa  too  larga  to  ba 
antiraly  included  in  one  axposura  af  filmad 
beginning  in  tha  uppar  Irft  hand  corner,  left  to 
right  and  top  to  bottom,  as  msny  frames  as 
required.  The  following  diagrams  illustrsta  the 
method: 


Las  cartas,  planches,  tableaux,  etc..  peuvent  etre 
filmOs  k  das  taux  da  reduction  diffSrents. 
Lorsque  le  document  est  trop  grsnd  pour  strs 
reproduit  en  un  soul  clichS,  il  est  films  S  partir 
da  Tangle  supOrieur  gauche,  de  gauche  4  droita. 
at  da  haut  an  baa.  an  prenant  la  nombre 
d'imegea  nOcassaira.  Las  diagrammas  suivsnts 
illustrant  la  mOthoda. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

'ij^rt.  ■:  jjL 


■ppapH 


MICROCOPY   RESOLUTION   TEST   CHART 

(ANSI  ond  ISO  TEST  CHART  No    2) 


1.0 


I.I 


1.25 


Hi 

ISO 


12.8 

■  3.6 


2.5 
2.2 

1.8 

1.6 


^  APPLIED  IN/MGE     Inc 

^^  '653   tost   Ma.n   Slresl 

"^SS  Rochester,   New   York        14609       USA 

•-^  (716)   482  -  C300  -  Phone 

^1=  (716)   288  -  5989  -  Fo, 


A  IMIAC  TIC  AL  TREATISE 

THE  STKKL  SQUARE 

ASU 

Its  Api)li('ati()n  to  Everyday  Use 


ItKISG  A.V  K.XIIAISTIVK  (()I.I,K(  HON  OF  fiTKKI,  HHIVKK 
I'KOIILKMS  AND  SOI.I  TIONS,  "  Ol.D  AND  NEW,  "  WITH  VANV 
OKIOINAI.  AND  fSKKri.  ADDITIOXf*,  KOK.MINO  A  COMPLKTK 
KNCYCI.OI'KDIA  OK  STEKI-  SlilAKE  KNOWLEIKiK,  TOOETIIKK 
WITH  A  IIRIKK  msTOKY  OK  THE  HQIARE,  AND  nESCKIPTrON 
OF     TAIII.ES,    KEYS     AND    OTHEK     AIDS    AND    ATTACHMENTS 

REVISED    EDITION 
IS  TWO  VOLUMES 


HV 

FRED    T.    HODGSON 

Member  o(  Canadian  Association  of  Archltt^'ts,  Killtor  of  "Xatloiiul 

Bulliler,"  Author  of  "MoiliTiit'ariieiitry.  "  "('nmmoiiSciisi  lluiiil- 

railinK"    and   other   prai'tUal    works   on   Hiillilinit,    ctr. 


VOLUME  II 


CniCAGO 
FREDERICK  J.  DRAKE  &  CO.,  PUBLISHERS 


l!K«t 


LaiXlUil  J9«Oo.    ^d.t.»,A«r 


CoMVmuHT  liH.'t 

HV 

V 

.(■•iiRK  K  J.  Drake  * 

I'HK'AIIO. 

COPVBIHHT   IMUU 
HY 

«'t». 

Fm(i>kruk  .r    Dkake  St 

Cii. 

C'HII'AHO. 

<K4ni!^:>8 


r»:«iM«C'- "  ■  -  ( -'>» 


PREFACE 

In  preparing  t'  s  second  volume  of  practical 
work  for  steel  '  jare  users,  it  was  itr'^Tative 
I  should,  in  some  measure,  give  ex?  '■■.'.  ■'%  and 
rules  that  were  somewhat  similar  to  number 
that  appeared  in  the  firsi  volume;  though  on 
close  comparison  it  will  be  found  that  the  simi- 
larity is  apparent  only,  as  different  solutions  of 
similar  problems  are  rendered. 

Perhaps  it  is  unnecessary  to  mention  this,  as 
the  expert  workman  will  recognize  the  differ- 
ence at  once;  but,  in  order  to  make  explanation 
beforehand  to  the  many  thousands  of  readers, 
who,  while  ot  being  pcrts  now  in  the  use  of 
the  square,  intend  to  'jcome  so  as  quickly  as 
possible,  I  thou  'hi  it  better  to  mention  the  mat- 
ter in  *]  's  prefa.  * 

The  ,  .  rchaser  of  this  volume  will  find  many 
things  in  it  that  are  original,  and  many  more 
that  have  been  culled  from  the  best  work  of 
experts  and  which  have  appeared  in  some  one 
or  other  of  the  great  number  of  trade  journals 
that  have  been  published  in  this  country,  in  Eng- 

3 


Km'^i^: 


E^-^- 


:^m!m 


^'T^rf^i^me'sm'^m^  t»  ^ 


4  PREFACE 

land  and  Australia,  during  the  last  twenty-five 
years.     As  before  stated,  there  are  a  number  of 
things  original  in  the  volume  that  have  never 
before  been   published,  some  of  my  own   and 
some  that  have  been  furnished  by  experts.     I 
have  made  it  my  business  to  write  to  every  per- 
son—at home  and  abroad— that  I  could  hear  of 
or  read  of,  who  had  made  a  study  of  the  steel 
square,  asking   for  anything   they   might   have 
that  was  new  and  useful  on  the  subject,  and  tell- 
ing them  I  was  preparing  a  new  and  exhaustive 
work  on  "The  Steel  Square."     While  all  did  not 
respond,  I   may  state  that  over  75  per  cent  of 
those  written  to  did;  and  while  the  great  major- 
ity had  nothing  to  offer,  a  groat  many  sent  me 
"cuttings"   from   journals,  on  the   subject,  with 
diagrams  and  suggestions.     Many,  of  course,  of 
the  problems  sent  me  I  could  not  make  use  of  for 
obvious  reasons,  while  several  gentlemen  sent 
me— for  publication— a  number  of  valuable  prob- 
lems and  solutions  which  I  have  embodied  in  the 
work    and    all    the   writers,   without   exception, 
wished   me   "God   speed"    in   the   work    I    had 
undertaken. 

It  is  now  in  order  for  me  to  publicly  thank 
those  who  have  so  kindly  aided  me  in  getting 
together    so    much    valuable   material    for    the 


'§ 

M 


PREFACE  5 

workman  as  will  be  found  between  the  covers  of 
these  two  volumes,  and  I  am  sure  my  readers 
feel  as  I  do  in  the  matter. 

Among  those  who  might  be  named  as  having 
aided  me  materially,  I  may  mention  Mr.  Woods, 
Mr.  Reissman,  Mr.  Stoddard,  and  Mr.  Penrose 
of  Trafford  Park,  England,  and  Mr.  Joseph  Wil- 
cox of  Sidney,  Australia,  to  whom  I  tender 
special  thanks. 

In  conclusion,  I  may  add,  it  is  the  intention  of 
the  publishers,  should  any  new  thing  arise  in 
connection  with  the  steel  square  and  its  applica 
tions,  to  have  the  same  embodied  in  the  present 
volumes,  or  issue  a  supplementary  volume  if 
necessary. 

If  by  these  volumes  I  have  been  instrumental 
in  aiding  and  assisting  the  operative  workman 
to  earn  a  little  more  wages  than  formerly  he 
received,  or  have  helped  him  to  better  his  condi- 
tion in  any  way,  it  will  be  a  gratification  and 
pleasure  to  me,  and  will,  in  some  measure,  repay 
me  for  all  the  effort  I  have  expended  in  bringing 
together  the  matter  contained  in  them. 

Fred  T.  Hudg.son. 

Collingwood,  Ont.,  1903. 


ft 


PREFACE  TO  SECOND  EDITION. 

VOLUME  II. 

In  this  volume,  as  in  the  previous  one,  there  are 
mtroduced  some  new  features  which  I  feel  con- 
vmced  will  prove  very  acceptable  to  the  practical 
student  and  well  repay  him  for  any  extra  trouble 
in  readmg  or  extra  cost  in  procuring.  A  second 
preface  to  this  volume,  further  than  a  simple  an- 

Vol.  11  of    The  Steel  Square  and  Its  Uses,"'  would 
seem  an  unnecessary  labor,  as  the  long  preface  in 
the  volume  of  the  first  edition  covers^  the  enti,^" 
ground  and.  as  a  rule,  the  operative  workman  does 
not  take  much  stock  in  a  long-winded  preamble. 
It  ,s     meat,     not  preliminaries,  the  active  work- 
man looks  for,  and  m  this    olume  he  will  find  plenty 
of  the  former  and  as  few  as  possible  of  the  latter 
1  am  convmced  in  my  own  mind,  that  the  two 
volumes  of  th,s  edition  oflfer  to  the  reader  nearlv 
double  the  amount  of  good,  honest  and  useful  mate- 
ral,  than  can  be  found  in  any  other  similar  work. 

nrnhl.n      ^'   ^''''   '"'^'^   ^"""^^   Solutions  of  useless 
problems   as  every  example  ofifered  is  a  tried  one 
and  one  hkely  to  confront  the  workman  at  anv  hou; 

sale  of  this  work  may  be  attril,uted.  While  it  is 
hardly  fair  to  expect  for  this  edition,  the  same  grat- 
ifymg  results  that  followed  the  first.  I  feel  svitis- 
fied  the  publishers,  as  well  as  nvself.  will  ha^^  i^o 
reason  to  regret  the -issuing  of  his  second  enlarged 
and  improved  edition.  ^ 

r  u-  ,    T  .  ^^^°-  T.  Hodgson. 

Lollingwood.  July,  1909. 

6 


THE  STEEL   SQUARE 


?—  tSSm  i»'      —     •/•*•     ».••■      fc***     A^**     — ••••     ^MM     _••••     «••"    ■  _«••  _  aaMit     _«■     ■■■■'■     ■^■*     ■ 


bi: 


U^M' 


'Q\.!'u'\.,miT^-\.!i7ii^7^^^^^^^ 


ANOTHER  STEEL  SQUARE 

Since  I  wrote  up  a  description 
of  the  various  squares  in  the  first 
volume  of  this  work,  I  have  been 
favored  by  a  correspondent  with 
the  following  description  of  a 
square  which  is  the  invention  of  a 
man  in  Augusta,  Maine.  I  have 
not  seen  the  square,  nor  have  I  seen 
a  good  illustration  of  it,  but  my 
correspondent  says  of  it,  "It  is 
an  ingenious  tool  for  framing  and 
can  be  adjusted  so  that  work  may 
be  rapidly  and  accurately  laid  out, 
and  it  is  capable  of  adaptation  to 
all  classes  of  buildings  where  wood- 
en frames  are  used.  I  have  used 
one  for  a  number  of  years,  and 


ij^T|Tj;r['ip|i;^T|r^Tp^T|T^r|T|T|r|;r|T^r|r|;iVi'|;i'|T|Tri 


TTT 


m 


'titi 


8 


PRACTICAL    USES  OP 


find  u  quite  handy  for  all  kinds  of  work  where 
fmber    ,s    employed."      The    description    »ent 
3  as  follows:   The  body  of  the  square  has  two 
tongues    arranged    „po„    it    a.    right    angles. 
Three  bars  are  so  secured  to  the  tongues  that  by 
loosenmg  set-screws  at  the  ends  they  may  be 
placed  at  any  points  desired,  the  bolts  sliding  in 
slo  s  cut  m  the  tongues.     One  of  these  bar,  has 
a  slot  cut  ,n  it  the  greater  part  of  its  length,  on 
wh,ch  two  gauge  blocks  are  secured  by  thumb- 

forth  and  adjusted  m  any  position  desired.     To 

the  body  of  the  tool,  on  one  side,  is  secured  a 

gauge,  also  capable  of  adjustment  by  means  of 

crews.    A  bar,  which  is  secured  to  one  of  the 

ongues  and  to  the  slotted  cross-bar,  is  used  in 

fratnmg  roof  timbers  and  the  like  where  diag- 

onals  are  to  be  cut. 

It  will  be  impossible  to  explain  all   the  ways 
.n  which  the  tool  may  be  used.     In  laying  out 

sTde  iToi    ,  '""T  °"  '""""■  "^^  ^-^'^  °"  -e 

of  the  "i,"'"'  "•"  ''"""'  ^''^•^  °f  'he  body 

of  the  square  will  come  at  the  outer  edge  of  the 

mortise  and    there    secured.      A    bar    is    then 

adjusted  so  that  the  space  between  it  and  the 

sin 'l    "     .  t  ""  "'■""'   °'  '"^   -"«-■      For 
single  work  this  is  all  that  is  necessary  for  mark- 


THE  STEEL  SQUARE  9 

ing  the  sides  of  the  mortises  and  tenons.  For 
double  work  the  second  mortise  is  made  by  the 
adjustmer*-  of  the  other  two  bars,  so  that  all 
change  of  the  tool  or  liability  to  error  is  avoided. 
To  mark  the  ends  of  mortises  and  tenons,  the 
end  gauge  is  adjusted  at  the  right  distance  from 
the  tongue  lying  parallel  with  it,  the  tool  moved 
alor  T  on  the  inside  of  the  gauge,  and  outside  of 
the  tongue  the  ma'king  is  complete.  For  framing 
roof  timbers  the  L  r  for  this  purpose  is  adjusted 
at  the  desired  angle,  and  by  this  the  Lead  and 
foot  of  rafters  or  braces  can  be  marked,  without 
changing  the  tool,  the  marks  for  the  pitch  of 
roof  being  put  upon  the  slotted  cross-bar  and 
end  tongue.  The  two  small  gauges  are  espe- 
cially designed  for  use  in  cutting  gains  for  shelv- 
ing, and  being  adjusted  at  the  proper  places  on 
the  cross-piece  and  secured,  proper  measure- 
ment will  be  given. 

As  I  stated  before,  I  have  not  seen  this  square 
and  am  obliged  to  take  the  foregoing  as  being 
correct,  but  it  seems  to  me  that  if  t.'ic  square  is 
as  useful  as  outlined  it  ought  to  be  better  -mown, 
but,  ha  ;^ing  inquired  at  several  large  h  ware 
houses  regarding  it,  I  find  it  entirely  unknown 
among  dealers.  If  any  of  my  readers  know  any- 
thing of  this  square,  as  to  its  usefulness,  price, 


to 


PRACTICAL  USES  op 


where  it  may  be  obtained,  and  will  acJvise  the  oub 
of  .t  .s  made  at  greater  len,  h  n.  future  editions 

COLOR    OF  SQUARES 

It  is  only  a  ,W  years  a^o  since  any  changes 

^quare  that  has  been  in  my  possession  for  fifty 

steel  surface,  bur  to-day  it  is  oxi.lize.l  and  cov- 
ered  w,th  a  coating  that  protects  it  from  rust  and 
f  cm  the  weather.    This  happens  to  all  po  Lhed 

with   mo,st   fingers.     This  condition,  however 

legible,  indeed,  ,n  many  cases  the  figures  and 
mes  become  almost  invisible,  a  very  gr'eat  ob^ec 

whe;e  the  fi"'     r""  '"•""  -^>'-™"s  cases 
where  the  figures  have  been  mistaken  and  tin-. 

tt:x::'°"^'"'°''^^°"'"— -■-■^ 

A  polished  square  should  never  be  rubbed  with 

su^chYbr^  "'  """^  '""'  ^"''^•---     While 
such  r^bUne  may  make  the  sides  of  the  square 
look  bright  and  "tidy  "  it  i,  «„„  ,„  •  ■         ■ 
squa  e.  efface  some  of  the  figures  and  markings, 
and  leave  the  surface  more  susceptible  to  rul 


THE  STEEL  SQUARE 


II 


A  little  neat  oil  applied  once  in  a  while  and 
rubbed  with  a  soft  rag  is  the  best  way  to  treat  a 
polished  square. 

Since  nickel-pL-ting  came  in  use,  many  squares 
are  so  treated,  and  this  has  many  advantages,  as 
under  ordinary'  circumstances  there  will  be  no 
rusting  and  the  square  will  always  be  bright  and 
"tidy"  looking.  Under  a  bright  sun  It  is  often 
difficult  to  read  the  figures  or  find  the  lines 
required,  and,  if  the  tool  is  exposed  to  the  sun's 
rays  for  a  short  time,  it  will  become  so  uot  as  to 
preclude  handling.  A  careful  workman,  how- 
ever, will  not  allow  his  square — or  any  of  his 
tools,  for  that  matter — to  be  exposed  to  the  sun 
for  any  length  of  time,  s  a  two-foot  blade  often 
expands  as  much  as  one-eighth  of  an  inch  in 
length,  when  made  hot — a  condition  the  good 
workman  cannot  allow. 

Squares  electroplated  with  copper,  in  my  opin- 
ion, are  much  better  than  either  polished  steel  or 
nickel-plated  ones.  The  copper  color  is  not  so 
severe  on  the  eyes,  and  the  shadows  cast  by  the 
cuttings  in  the  figures  and  markings  bring  up  the 
figures  almost  in  relief,  so  that  they  arc  readily 
seen.  Age  gives  a  copper-plated  square  a  fine 
antique  color  which  is  restful  to  the  eye  and  a 
protection  to  the  square. 


la 


PRACTICAL  UvSES  OF 


If  the  square  is  allowed  to  oxidize,  and  is  then 
polished  on  the  surface,  we  get  a  fme  copper  fin- 
ish on  the  tool  with  dark  oxichze.l  figures  and 
markings. 

Best  of  all  is  the  blued  or  what  may  be  term.d 
the    "gun   metal"    square.      These   squares   are 
oxidized  or  blued  by  some  process  unknown  to 
me,  and  the  figures  and  lines  are  filh^d  in  with 
some  kind  of  white   enamel,  that  brings  them 
out  m  great  shape.     A  very  handsome  blued  and 
enameled   crenelated   square   was   sent    me   for 
examination    by  The    Peck,  Stov/   and    Wilcox 
Company,   of  Southington,    Conn.,    which    lies 
before  me  as  I  write;   and  it  seems  to  me  that 
this  method  of  bluing,  and  white-enameling  the 
figures,  is   one    of   the    greatest   improvements 
made  on  the  square  for  many  years.     To  keep 
the  copper,  nickel-plated  or  blued  squares  in  nice 
order,  they  should  be  rubbed  over  once  in   a 
while  with  good  machine  oil  applied  with  a  soft 
rag.     By  this  simple  process  a  square  may  be 
kept  new   m  appearance  for  a  long  period  of 
years. 

These  few  hints  on  the  qualities  of  squares 
and  their  care  will,  doubtless,  prove  of  value  to 
my  readers. 


THE  STEEL  SQUARE 


«J 


SOME   ODD    PROBLEMS 

The  workman  often  is  confronted  with  very 
curious  problems,  and  is  as  often  put  to  his  wits' 
end  to  solve  them.  Here  is  a  problem  and  its 
solution  that  will  prove  interesting,  though  it  is 
not  likely  :<>  \»:  met  very  often: 

Suppose  a  box  twelve  inches  square  to  be  set 


Fii..  I 

on  the  ridge  of  a  roof,  as  shown  in  Fig.  i.  The 
roof  is  half  pitch,  or  rises  12  inches  to  the  foot. 
Required,  the  cuts  for  each  side.  The  corner 
X  is  7}<  inches  from  the  ridge;  therefore  it  is 
7j4  inches  lower  than  H.    The  cut  for  side  A 


14 


PRACTICAL  USES  OP 


Is  therefore  VA  to  12.  X  is  also  VA  inches  lowtr 
than  Y.  The  cut  for  H  is  7^  to  q%.  The  cuts 
for  the  other  i,icle  are  found  in  the  same  way. 


PlO.  3 


For  proof  see  Fig.  2.    The  side  of  the  box 
being  12  inches,  therefore,  when  set  on  the  roof 
with  one  corner  touching  the  ridge,  or  in  any 
other  position,   it   will   reach    12    inches    hori- 
zontally  over  the  roof.    And  with  a  run  of  12 
inches  you  have  a  rise,  or  rather  a  fall  in  this 
instance,  corresponding  to  the  pitch  of  your  roof. 
With  the  run  on  one  side  of  your  square  and  the 
rise  or  fall  on  the  other,  you  will  get  the  tonect 
cut  eveiy  time. 


THE  STEEL  SQUARE 


If 


Two  other  problems  are  given  here  which  I 
am  sure  will  prove  interestinjjj: 

To  determine  by  a  steel  square  the  result  of 
any  number,  for  example,  6,  multiplied  by  the 
sin  45  \  Take  6  on  both  blade  and  tongue,  and 
mark  the  line  ac^f)c=f>.  Then  <:</ drawn  to  the 
midd!eofa<5-6xsin45  .   Sin  75  =0.7071 1.  Fig. 3. 


/V<?^/<-/;/.  -Referring  to  any  rectangular  frame- 
work which  slopes  alike  on  all  four  sides.  Given— 
The  rise  and  seat  of  the  corner  posts.  Required— 
The  cuts  for  the  ends  of  the  corner  posts,  the 
blade  of  the  bevel  to  be  applied  to  the  two  faces 
adjoining  to  the  ridge  line. 

By  the  Steel  Square.— On  the  blade  take  the 
rise.  On  the  tongue  take  the  seat  then  sin 
45'-  Mark  along  the  tongue.  The  angle  made 
by  the  tongue  and  the  line  is  the  proper  bevel. 

Prodlem.— Given— Thit  run  and  rise  of  a  com- 


m 


I« 


PR'-TICAL  l^SES  OF 


mon  roof.  Wo  wish  to  i)lacc  upon  it  a  perpen- 
dicular  square  pipe  t..  stand  upon  the  roof  dia- 
mond-shape.  thus:  <ft>  AVy/.m-^-The  cuts  for 
the  bottom  end  cf  th<^  pipe;  the  blade  of  the 
bevel  to  b<;  a|,plie<l  to  the  two  faces  adjoining,  the 
lowermost  vertical  edge. 

nylhc  Steel  Squarc.~On  the  blad.-  take  the 
run  of  the  ro^f.  On  the  tongue  take  the  rise 
-<  s>n  45°.  Mark  alonK'  the  tongue.  The  angle 
made  by  the  tongue  and  the  line  is  the  proper 
bevel. 

If  the  bevel  is  applied  to  mark  the  end  of  the 
higher  halt  of  the  pipe,  we  have  on!  to  reverse 
the  direction  of  the  stock  of  the  bevel  so  as  to 
make  an  obtuse  angle. 

These  two  problems  are  very  much  alike. 
Their  demonstration  is  a  good  study  in  solid 
geometry. 

Suppose  we  wi.h  to  cut  an  opening  in  a  roof 
for  a  round  pipe  or  a  tile,  so  that  the  pipe  or  tile 
will  stand  vertical  through  the  opening.  The 
exact  form  of  the  opening  may  be  obtained  by 
the  following  method,  which  is  taken  from  "Car- 
pentry  and  Building"  of  New  York:  set  the  pipe 
or  tile  to  be  used  on  the  roof  at  the  point  where 
the  hole  is  to  be  cut.  as  shown  at  Fig.  4;   plumb 


THE  STEEL  SQUARE  ,7 

it  with   a  level  or   plumb   and   place  a  square 
alongside  of  it  as  is  shown  in  the  sketch.     Keep. 


Fig. 


ing  the  blade  of  the  square  in  contact  with  the 
pipe,  move  it  around  ^he  circumference  of  the 
pipe,  touching  the  roof  at  points  about  one  inch 
apart  anc  makinjr  a  mark  on  the  roof  at  each 
point  of  contact.  When  the  entire  circumfer- 
ence of  the  pipe  has  been  traveled,  join  the 
points  marked  on  the  roof,  and  the  figure  out- 
lined will  be  a  perfect  t^liipse  and  of  exactly  the 
size  required.     Nothing  remains  but  to  cut  the 


z8 


PRACTICAL  USES  OF 


I 


hole,  and  if  this  is  done  correctly  the  pipe  will  be 
found  to  fit  exactly.  This  method  is  simple  and 
perfectly  accurate,  and  as  it  requires  no  calcula- 
tion, it  can  be  done  by  any  workman  who  can 
handle  the  tools  and  pipe. 

Often  men  working  in  wood-working  factories 
find  it  necessary  to  increase  or  decrease  the 
speed  of  a  shaft  a  few  revolutions.  To  get  just 
the  size  of  pulley  necessary  for  this  increase  or 
reduction  of  speed  requires  close  calculation 
when  figures  are  used,  but  a  square  will  do  it 
off-hand  and  correctly  every  time.  For  instance, 
it  has  been  ascertained  that  a  pulley  20  inches  in 
diameter  on  a  certain  shaft  will  give  a  speed  of 
13  revolutions  per  second.  If  it  be  desired  to 
reduce  the  speed  to  ii>^  revolutions  per  second, 
the  20-inch  pulley  must  be  removed  and  one  of 
different  size  substituted. 

To  solve  this  problem  with  a  steel  square,  take 
20  inches  on  the  tongue  and  13  inches  on  the 
blade.  Place  both  of  the  marks  on  the  edge  of 
a  straight  board  and  draw  a  line  parallel  with 
the  blade.  Move  the  square  parallel  with  this 
line  until  the  tongue  shows  iij^  inches  instead 
of  13.  Notice  the  reading  on  the  blade  and 
edge  of  the  board.  This  reading  will  be  size  of 
pulley  required  in  place  of  the  20-inch  concern, 


THE  STEEL  SQUARE  19 

necessary  to  give  the  desired  speed  oi  \i]4  revo- 
lutions per  second.  The  principles  involved  in 
the  calculations  here  made  have  been  amplified 
and  put  into  a  more  convenient  shape  for  every- 
day use  in  the  slide  rule. 

In  this  perfected  form  some  mechanics  are 
familiar  with  the  tool,  but  all  do  not  recognize  it 
in  the  slide  rule.  The  principle  upon  which  all 
slide  rules  work  is  that  of  the  square  and  the 
piece  of  board  above  mentioned.  There  are 
several  more  problems  of  this  kind  that  might 
be  described  with  interest  to  the  reader. 

To  find  the  number  of  cogs  in  a  wheel,  pitch 
of  cogs  and  diameter  of  wheel  given,  set  the 
bevel  to  the  pitch  on  the  tongue  and  3.14  on  the 
blade. 

Given  the  diameter  of  a  wheel  to  pitch  line, 
and  number  of  cogs,  to  find  pitch  of  cogs.  A 
wheel  70  inches  diameter  has  146  cogs;  what  is 
their  pitch?  146  inches  being  too  great  to  set  on 
the  square,  we  take  proportional  parts,  setting 
the  bevel  to  V  inches  or  83^  inches  on  tongue, 
and  4«  inches  or  \^%  inches  on  blades.  Tighten 
the  screw  of  fence,  and  move  the  bevel  to  3.14  on 
blade,  and  the  number  given  on  the  tongue 
multiplied  by  8  will  be  the  required  pitch. 

li.  we  wish  to  divide  a  circle  into  a  given  num- 


ao 


PRACTICAL  USES  OF 


ber  of  parts,  we  proceed  as  follows:  Multiply  the 
radius  by  the  corresponding  number  in  column 
A,  as  per  table,  and  the  product  is  the  chord  to 
lay  off  on  the  circumference. 

Given  the  side  of  a  polygon,  to  find  the  radius 
of  the  circumscribing  circle.  This  problem  has 
previously  been  tabulated,  but  by  multiplying 
the  given  side  by  the  number  corresponding  to 
the  polygon  in  column  B,  in  most  cases  we  obtain 
the  answer  more  expeditiously. 


No.  of  sides 

A 

or  parts 

A 

B 

3    • 

.    Triangle 

1.732 

•5773 

4    • 

.     .     Square 

I.414 

.7071 

5    • 

.     Pentagon 

1. 175 

.8006 

6    . 

.     Hexagon 

radius 

side 

7     • 

.     Heptagon 

.8677 

1-152 

8    . 

.     Octagon 

•7653 

1.3065 

9     • 

.     Nonagon 

.6840 

1.4619 

ID      . 

.     Decagon 

.6180 

1.6180 

II       . 

.     Undecagon 

•5634 

1-7747 

12      . 

Dodecagon 

•5176 

1.9318 

Given  the  iHamclcr  of  tx  circle,  to  find  the  side  of 
a  squdjc  of  equal  ana. S^X.  a  bevel  to  q}i  on  the 
tongue  and  1 1  inches  on  the  blade.  Then  move 
the  bevel  to  the  diameter  of  the  circle  on  the 
blade  and  the  tongue  gives  the  answer.  When 
the  circumference  is  ^n'vtn  instead  of  the  diam- 


THE  STEEL  SQUARE 


ai 


eter,  set  the  bevel  to  s^A  inches  on  tongue  and 
I9>^  inches  on  the  blade. 

To  find  the  number  of  square  yards  in  a  given 
area  we  must  proceed  as  follows:  These  prob- 
lems require  a  bevel  of  9  on  tongue,  and  the 
length  or  width  of  the  surface  to  be  measured  on 
the  blade.  The  bevel  is  then  moved  to  the 
remaining  dimension  of  the  area  on  the  tongue, 
and  the  number  on  the  blade  indicates  the  square 
yards  contained. 

If  the  diameter  of  a  circle  is  given,  we  can 
determine  the  circumference  as  follows:  Set  a 
bevel  to  7  on  tongue  and  22  on  blade;  the 
answer  will  be  found  on  the  latter.  In  every  case 
when  a  reverse  problem  is  presented  the  bevel 
will  solve  it  unchanged;  we  merely  look  for  the 
answer  on  the  other  blade  of  the  square.  For 
instance,  if,  after  sobirg  the  above,  we  are 
required  to  determine  the  diameter  from  the 
circumference,  we  still  use  the  same  bevel. 


SOME    DIFFICULT    PROBLEMS    AND    THEIR    SOLUTION 

I  am  indebted  to  Mr.  Fred  La-^y  of  San 
Francisco,  Cal.,  for  some  of  the  following  inter- 
esting problems  and  their  solutions.  They 
make  excellent  practice  for  the  young  student 
who  has  made  up  his  mind  to  learn  all  possi- 


32 


PRACTICAL  USES  OF 


ble  concerning  the  square  and  what  may  be  done 
with  it. 

The  workman  is  often  confronted  with  prob- 
lems in  oblique  framing  that  are  difficult  to  solve 
unless  he  possesses  knowledge  of  a  high  order  of 
solid  geometry.  *See  how  Mr.  Lascy  handles  the 
square   first   in  splayed  work,  then   in   oblique 
bevels.    To  construct  on  a  base  of  any  number 
of  salient  corners,  a  solid  in  which  every  two 
adjoin'ng  faces  slope  together  toward  the  hori- 
zon   form;:  :  a  hip, 
and  the  same  rise,  of 
any  lengths.     To  de- 
termine by  the  steel 
square  on  a  line  all 
the   angles  that  can 
be  required  for  such 
constructions,   fi    st 
draw  to  a   proper 
scale  the  run  A  B,  Fig. 
5.  the  rise    BD,  and 
the  slope  AD.    This 
triangle  is  supposed 
^'  ^  to  stand  perpendicu- 

lar to  the  plane  of  the  paper.  Construct  the 
angle  ABC=B=any  angle  whatever.  If  all  the 
corners  of  the  base  arc  alike  in  degrees,  B=i8o 


THE  STEEL  SQUARE  ,3 

divided  by  the  number  of  corners.  If  the  corners 
are  unlike  in  degrees,  consider  each  corner  by 
itself  and  make  6=90°,  minus  half  the  degree,  of 
this  corner.  Draw  AC  and  BC,  which  is  always 
the  seat  line  of_'the  hip.  ABC  is  always  half  the 
corner  of  the  base.  All  lines  drawn  from  the  run 
to  the  seat  line  must  be  at  right  angles  to  the 
run. 

By  the  steel  square.  Place  the  slope  on  the 
blade.  Place  AC  (run  tan  B)  on  the  tongue. 
Mark  along  the  blade  for  the  face  cut  against  the 
hip  line  of  boards  which  have  the  direction  of 
jack  rafters.  Mark  along  the  tongue  for  the  face 
cut  against  the  hip  line  of  horizontal  boards;  for 
the  top  cut  of  purlins;  for  ihe  top  cut  of  a  miter 
box  to  miter  the  horizontal  boards. 

By  the  steel  square.  Place  the  slope  on  the 
blade.  Place  EF  (rise  tan  B)  on  the  tongue. 
Mark  along  the  tonguv.  for  the  miter  cut  across 
the  square  edp  of  horizontal  boards;  f  jr  the 
down  cut  of  jrlin;    for  the  down  cut  of  a 

miter-box  to  m.  -r  the  horizontal  boards,  lying 
flat  in  the  box.  If  the  boards  be  not  mitered,  we 
require  the  butt  cut  across  the  square  edge  of 
boards.  For  this  draw  BG,  at  right  angles  to 
the  slope;  make  BK=AG,  and  draw  KI=run 
cosine  tan  B. 


m 


»4 


PRACTICAL  USES  OF 


.  i 


By  the  steel  square.     Place  the  rise  on  the 
blade.     Place  KI  on  the  tongue,  and  mark  along 
the  tongue  for  the  'ntt  cut  across  the  square 
edge.     Make  BH=seat,  then  HD=hip  line  with 
its  top  and  bottom  bevels.     For  the  diedral  miter 
at  right  angles  to  the  hip  line,  which  is  half  the 
angle  between  any  two  adjoining  sloping  faces. 
By  the  steel  square.     Place  the  hip  line  on  the 
blade.     Place  EF  (rise  tan  B)  on  the  tongue, 
and  mark  along  the  tongue  for  the  diedral  miter, 
To  back  a  hip  rafter  by  a  gauge  mark,  make 
B«  =  half  the  thickness  of  the  hip,  and  draw  nm 
to  tht  seat;  from  the  toe  of  the  hip  rafter  make 
Hp=um,  and    from  p  draw  the    gauge    mark 
parallel   to  the  hip.      For  a  four-sided   hipped 
roof,  we  may  need  the  side  cut  across  the  top 
square  edge  of  the  hip  rafter  against  the  ridge. 
For  this  make  Bv=ha\i  the  seat,  and  draw  v^e  at 
right  angles  to  the  seat,  to  meet  bu  at  right 
angles  to  the  run. 

By  the  steel  square.  Place  QUV  (seat  cotan  B) 
on  the  tongue;  place  the  hip  line  on  the  blade 
and  mark  along  the  blade  for  the  side  cut,  short- 
ening  the  hip  according  to  the  half-thickness  of 
the  ridge,  thus:  It  B«  =  half-thickness  of  the 
ridge,  draw  a  distance  Bw  parallel  to  the  rise; 
where  this  line  cuts  the  hip  shows  the  shorten- 


THE  STEEL  SyUAKE 


2S 


-if. 


ing.  For  a  square  corner,  6=45",  tan  F{= co- 
tan  B=i.  In  this  case  tangents  of  B  may  be 
drawn  or  not. 

Again,  let  the  raking  molding  of  a  pediment  = 
AD,  which  is  to  miter  against,  as  horizontal 
molding  that  forms  with  the  run  of  the  pediment 
any  angles  whatever.  8-90°  minus  half  of  this 
angle.  The  raking  molding  must  be  placed  in 
the  miter-box  with  that  part  of  the  molding 
which  is  vertical  when  in  position  against  the 
side  of  the  box. 

Lay  the  steel  square  on  top  of  the  box.  Place 
AG  (run  cosine)  on  the  blade;  place  AC  (run 
tan  B)  on  the  tongue,  and  mark  along  the 
tongue  for  the  top  cut  of  the  miter-box.  Again, 
place  the  run  on  the  blade;  place  the  rise  on  the 
tongue  and  mark  along  the  tonj,ue  for  the  down 
cut  of  the  miter  box. 

The  miter  of  the  horizontal  molding  equals 
one-half  the  corner  on  the  ground  plan.  If  the 
diagram  ADB  equals  half  the  gable  end  of  a 
rectangular  building,  to  miter  the  raking  plan- 
ceer  against  the  horizontal  planceer,  which  slopes 
as  the  roof  and  runs  along  the  eaves  at  right 
angles  to  the  run  of  the  gable,  both  planceers 
being  in  the  same  plane. 

By  the  steel  square.     xMace  the  slope  AD  on 


a« 


PRACTICAL  USES  OF 


the  blade;  place  the  run  AB  on  the  tongue; 
mark  along  the  blade  for  the  raking  miter;  ma  k 
along  the  tongue  for  the  horizontal  miter.  The 
horizontal  planceer  requires  the  wider  board. 

In  laying  off  angles  for  splayed  work,  lines  as 
long  as  possible  and  as  few  as  possible  are  the 
essentials  of  accuracy  and  comprehension  of  the 
subject. 

Another  problem  with  its  solutions  follows: 
Given  a  hopper  or  hipped  roof  which  stands  on 
a  base  whose  corners  have  any  angle  whatever, 
and  whose  sloping  sides  have  the  same  run  and 
the  same  rise  of  any  length.     Required  the  die- 

dral  angle  between  any 
two  adjoining    sloping 
sides  without  using  the 
hip  line;  or,  what  is  the 
same  thing,  to  find  the 
backing  of  the  hip  raft- 
er without    using    the 
hip  line.     First  lay  off 
AB=run,  Fig.  6,  BD= 
rise,  A  D  =  slope  line. 
Draw  ACat  right  angles 
to  the  run,  and  lay  off 
angle  ABC=9o°,  minus  half  the  angle  of  the  cor- 
ner of  the  base.    Angle  6 = half  angle  of  corner  of 


THE  STEEL  SQUARE 


'7 


base.  Make  BE = rise,  and  draw  EF  at  right  an- 
gles to  BC,  the  seat  of  the  hip  line.  On  the  edge 
of  a  board  lay  off  t 
GF,  Fig.  7,=slope 
AD.     By  the  steel 

square  on  the  lineT ^*    **^ 

GF,  place  EF  on  pio.  7 

the  tongue  and  apply  the  tongue  as  shown  on  the 
diagram,  moving  the  blade  until  the  blade  cuts 
the  point  G.  Mark  along  the  tongue.  EFG= 
one-hatf  the  required  diedral  angle=diedral 
miter.  By  calculation:  Cos  diedral  miter=rise 
cos  half-corner  angle  C.  The  demonstration 
may  be  studied  in  any  book  on  solid  trigonom- 
etry which  treats  on  the  right  triangular  pyra- 
mid. It  is  too  long  to  be  given  here.  Without 
using  the  steel  square,  draw  a  semicircle  on 
FG,  and  taking  FE  in  the  compasses,  mark  the 
chord  line  EF.  This  gives  the  diedral  miter 
EFG  to  back  the  hip  rafter. 

The  following  problems  in  oblique  bevels  will 
prove  both  useful  and  instructive  as  well  as 
interesting  to  the  studious  young  workman: 
Given  the  run  AC,  the  rise  CB,  Fig.  8,  and  the 
slope  length  BA  of  a  stick  of  rectangular  timber, 
which  butts  obliquely  at  the  upper  end  against  a 
vertical   plane  of  indefinite  length,  and  whose 


38 


PRACTICAL  USES  OF 


i 


seat  line=CE.     Angle  between  this  seat  and  the 
given  run  =  W,  which  may  be  any  angle  what- 
ever.    Required  the  side 
bevel  on  the  top  face  of 
the  stick  at  the  point  B  to 
fit  this  end   of  the  -.tick 
against  the  vertical  plane. 
First  draw  AE  at   right 
angles  to  the  run,  make 
AD^«1opeand  draw  DE. 
The   required  side  bevel 
=:EDA.    Demonstration: 
Let  the  triangle  EDA  re- 
volve   on   AE    as    on   a 
^k  hinge;  when  the  vertex  D 
reaches   tue    required 
height   of    the    rise,   the 
triangle  EDA  will  stand 
directly  over  the  triangle 
EGA,  the    line   ED   will 
coincide  with  the  vertical  plane,   the  line    DA 
will    coincide   with  the    top  center  line  of  the 

stick.     By  calculation,  tan  side  Level  =  -  °  ^^"  ^ 
,  ,,r  -  slope 

when  \V  ^  45  ,  tan  VV  ^  unity.  By  the  steel 
square  on  a  line:  Place  AE  (run  tan  W)  on 
the  blade,  place  the  slope  line  on  the  tongue, 
mark   along    the    tongue    for   the    side    bevel. 


Fig.  8 


THE  STEEL  SQUARE 


»9 


more 


When  W  is  an  < 

ccnvenient  to  ob         v,.  .»*^  i 

I  following  manner:  From  /  "he  center  point 
I  of  the  run,  draw  fg  at  right  angles  to  the  run. 
::    then  AD=twice/^.      If  the  stick  should   at  its 

lower  end  butt  obliquely  against  a  vertical  plane. 

the  side  bevel  is  obtained  in  precisely  the  same 

way  as  given.     By  this  method  we  may  obtain 

the  side  bevel  at  the  lower  end  of  certain  jack 

rafters,  making  planers    and   rafting  moldings. 

The  other  bevels  at  the  top  and  bottom  ot  the 

stick  do  not  require  any  remarks. 
Again,  given  the  y: 

run  KV,  Fig.  9,  the 

rise   KA    and   the 

slope  A V  of  a  stick 
of  rectangular  tim- 
■     berwhichatitslow- 
ered  end   V  butts 
obliquely  against  a 
vertical  plane  whose  ground  line  is  in  the  direc 
fon  of  VC      The  corner   K\C  =  .    n.ay  be  of 
any  size,  and  is  supposed  to  stand  perpendicu- 
lar to  the   plane  of  the  paper.     Required  for 
the  lower  end  of  the  stick  the  miter  cut  across 
the  top  face;    also  the  down  cut,  so  that  this  end 
of  the  stick  may  fit  against  the  vertical  plane 


30 


PRACTICAL  USES  OF 


): 


whose  ground  line  is  VC.  First,  from  the  pc'nt 
K,  and  at  right  angles  to  the  nin,  draw  I\C  to 
intersect  tht  ground  line  of  the  vertical  plane. 
From  the  point  A,  and  at  right  angles  to  the 
slope,  draw  AB  =  KC;  also  draw  BB;  then  AVB 
is  the  miter  cut  across  the  top  face.  Demon- 
stration: The  triangle  VKA  being  perpendicular 
to  the  plane  of  the  paper,  let  the  triangle  VAB 
revolve  on  VA  as  on  the  hinge,  until  the  line 
AB  comes  into  the  horizontal  position  over  KC; 
then  AVB  is  the  required  sloping  triangle  of 
which  KVC  is  the  plan.  By  the  steel  square  on 
a  line:  Place  KC  on  the  blade  and  the  slope 
VA  on  the  tongue.  Mark  along  the  tongue  for 
the  top  face  miter.  The  end  down  cut  may  be 
marked  by  a  bevel  set  to  i.he  angle  IIAV.  The 
stock  of  the  bevel  is  applied  along  the  bottom 
face  of  the  stick  with  the  blade  of  the  bevel 
pointing  upward  along  the  side  face.  The  most 
common  applications  are  the  rafters  which  butt 
obliquely  against  valleys,  hips  and  ridges.  If  the 
line  VA  represents  a  raking  planceer,  which  at 
the  lower  end  of  a  gable  miters  around  a  square 
corner  and  against  a  horizontal  planceer  that 
slopes  in  accord  with  VA,  then  the  angle  //  is 
always  45°,  and  CK  and  AB  will  each  equal  the 
run  VK.     The  miter  for  the  end  of  the  hori- 


THE  STEEL  SQUARE 


3« 


zontal  planceer  will  be  the  angle  ABV.  The 
horizontal  planceer  is  supposed  to  have  its  inside 
edge  beveled  to  fit  against  a  vertical  plane;  a 
square  mark  down  the  beveled  edge  is  the  down 
cut  through  the  thickness  of  the  mitered  end  of 
the  wider  horizontal  planceer.  VA  and  VK 
show  the  relation  between  the  width  of  the  two 
planceers.  VA  is  a  raking  molding,  mitering 

at  point  V  around  a  square  corner  and  against  a 
horizontal  molding.  The  lower  (.nd  of  the  ra- 
king molding,  mitering  at  point  Y  around  a  square 
corner  and  against  a  horizontal  molding.  The 
lower  end  of  the  raking  molding  should  be  cut  in 
a  miter-box,  with  that  part  of  the  molding  which 
is  nailed  against  the  gable  placed  against  the 
side  box.  The  foregoing  are  the  proper  cuts 
for  the  miter-box.  The  line  KC,  etc.,  may  be 
drawn  anywhere  along  the  line  KV. 

Here  is  an  excellent  graphical  method  of  find- 
ing the  areas  of  different  figures.  It  is  taken 
from  "The  American  Machinist,"  and  is  worthy, 
I  think,  of  a  place  in  this  work,  because  of  its 
compactness  and  simplicity. 

When  the  area  and  diameter  of  any  circle  is 
known,  by  this  method  the  area  due  to  any  other 
diam.eter,  or  a  diameter  due  to  any  other  area, 
may  be  determined.     Suppose  we  take  the  diam- 


PRACTICAL  USES  OF 

eter  2  wJth  the  area  3.1416  as  the  known  quantity 
from  which  to  calculate  all  others.  Any  other 
diameter  and  area  may  be  chosen,  but  this  one 
IS  the  easiest  to  remember.     Draw  the  indefinite 

straijrht  line  AB,  Fi^r. 
10,  and.  with  a  diam- 
eter equal  to  3.1416. 
draw    the    semi -cir- 
cumference     ADC. 
With  a  radius  equal 
to  2,  and   with  A  as 
a  center,  cut  the  semi-circumference  ADC  in  D. 
Throu^rh    D  erect  the   perpendicular   HH.  then 
wdl   the   distance  Ai{   he  a  constant   for  every 
diameter  and  area.     Let  it  be  required  to  find  by 
this  dia^rram  a  diameter  that  has  an  area  equal 
to  5.     Lay  off  AF^^.s  and  draw  the  semi-circum, 
ference  AGF,  then  will  the  distance  AG  be  the 
required  diameter.     If  the  (hameter  is  ^men  and 
the  area  required,  take  A  as  a  center,  and,  with 
the  given  diameter  as  a  radius,  cut  the  line  HF 
in  G.     Bisect  the  line  AG  at  right  angles  with 
the  hne.  cutting  AH   in   K,  then  will   K  be   the 
center  of  a  circle  passing  through  A  and  G,  and 
Its  diameter.  AF.  will  be  the  required  area. 

This  diagram  is  susceptible  of  a  great  variety 
of    applications.      The    diameters   on   the    line 


THE  STEEL  SQUARE  33 

AB  may   be   areas,   capacities,    weights,    tensile 
strengths,  or.  in  fact,  anything  that  is  made  up 
'   ^ir-a  anj   ,  constant  quah'ty.     The  distances 
ro-.n  A  .0  th  .  hne  H E  are  always  diameters,  sides 
^-^y  ^qu.re  or  some  constant  component  of  area 
The  distance  AE  is  different  for  each  kind  of 
proposn.on.  but  is  constant  for  every  proposition 
of  the  same  kmd.     If  we  say  that  a  bar  of  i-inch 
round   ,ron  has  a  safe  tensile  strength  of  ;  000 

lbs,  then  we  lay  off  AF-7  dr;,w  rh  7^ 

,.  ^         ^^  ~/,  draw  the  arc  on  th  s 

diameter  and  lay  off  AG  =  ,.    The  position  o 

G  g,ves  the  location  of  the  line  HE.  from  which 

he  ,ens,le  .trength  for  any  diam,-ter  may  be 

fo.,nd,      A    diameter    AD   will   have  a   .eLile 

strength  AC.  icnsiie 

The  advantage  of  thi.  diameter  lies  in  the 
fact  that  ,t  i»  impo^ible  to  remember  all  area. 
we,g  ts,  strength,  etc.,  while  it  is  comparati::,y 

,     ""'•   '-"°""'  't    -natters    no,   whether  the 

It  is  a  well-known  geometrical  fact  that  the 
angle  wuhin   a    semi-circumference   is  a     i'h! 

hi!  in  'T"  "'-'"'  -^y  -■=«  ='dvan  ag?o 
^h.s  tn  makmg  of  core-bo.es,  in   the  manner 


t-i. ' 


r 


34 


PRACTICAL  USES  OP 


shown  in  Fig.  it.     If  the  core-box  has  been  cut 
out    accurately,   then   the  square   will   touch    at 

three  places— th<  two 
edges  and  a  point  in 
the  curve.  If  it 
touches  at  only  two 
places,  one  being  on 
the  curve,  then  it  is 
not  cut  out  deep 
enough;  if  it  touches 
Fig.  II  only  at  the  two  edges, 

then  it  is  cut  too  deep  by  the  amount  of  clearance 
between  the  corner  and  the  curve.  By  giving 
the  square  an  oscillation,  to  make  the  corner 
sweep  the  entire  surface  of  the  curve,  the  accu- 
racy of  the  curve  at  that  point  may  be  ascer- 
tained, and  by  trying  at  several  points  the  truth 
of  the  whole  box  may  be  determined.  The 
square  may  thus  be  made  to  take  the  place  of  a 
templet  in  making  of  core-boxes,  with  the 
advantage  that  the  square  will  fit  any  size,  while 
a  templet  only  fits  one.  There  is  an  opportunity 
here  for  some  one  to  get  up  a  core-box  plane 
that  will  produce  any  semi-circumference  accu- 
rately and  quickly.  The  only  care  that  would 
have  to  be  exercised  by  the  workman  would  be 
the  placing  of  two  parallel  metal  strips  as  shown 


THE  STEEL  SQUARE 


35 


at  a  and  b.  These  strips  are,  of  course,  not 
necessary  when  simply  usin^  the  square  to  test 
the  accuracy  of  the  work,  but  would  be  necessary 
in  the  use  of  a  core-box  plane  made  on  this 
principle. 

If  we  wish  to  find  the  diagonal  of  a  square  or 
parallelogram,  all  we  have  to  do  is  to  set  the 
blade  of  a  bevel  to  8^  inches  on  the  tongue  and 
12^  inches  on  the  blade.  Then  screw  the  bevel 
fast;  and  supposing  the  side  of  the  square  in 
question  is  ii  inches,  move  blade  to  the  ii-inch 
mark  on  the  tongue,  keeping  blade  against  the 
square,  when  blade  will  touch  15  A  inches  on  the 
blade,  which  is  the  required  diagonal.  There  is 
no  special  reason  for  using  8->{  and  12^^;  other 
nur''^  rs  may  be  employed  provided  the  propor- 
tic  70  to  90  exists  between  them.     In  the 

pro  .v-in  just  solved,  as  in  all  that  follow,  the 
bevel  being  oncfe  set  to  solve  a  particular  ques- 
tion will  solve  all  the  others  of  the  same  kind, 
till  the  bevel  is  altered. 

To  find  the  circumference  of  an  ellipse  or  oval, 
we  jiroceed  as  follows:  Set  55^  inches  on  the 
tongue  and  8^4  inches  on  the  blade.  Then  set 
the  bevel  to  the  sum  of  the  longest  and  shortest 
diameters  of  the  ellipse  on  tongue,  and  the  blade 
gives  the  answer. 


•»^  • 


^f 


r 


«r 


36 


PRACTICAL  USES  OF 


If  it  is  desired  to  find  the  side  of  tlie  greatest 
square  which  ma>  be  inscribed  within  a  circle  we 
can  accomplish  it  with  the  aid  of  the  square  as 
follows:  The  diameter  of  a  circle  being  given, 
set  the  bevel  to  8^  inches  on  the  tongue  and  12 
inches  on  the  blade.  The  answer  will  be  found 
on  the  tongue. 

To  inscribe  three  small  circles  within  a  large 
circle  of  given  diameter,  set  to  6>^  inches  on 
tongue  and  14  inches  on  blade.  Move  the  bevel 
to  the  given  diameter  on  the  blade  and  the 
required  diameter  appears  on  the  tongue. 

Four  equal  circles  require  a  bevel  of  2.91  and 
14. 

Tfl  inscribe  polygons  zuithin  circles. — In  the  fol- 
lowing table,  set  the  bevel  to  the  pair  of  numbers 
under  the  polygon  to  be  inscribed: 
No.  i  f  sides    345       6        7     8      9     10    11    12 
Radius  s6  70  74     ''''f,    60  98   22    89   80   8s 

o.j  -'      '       '^ equal  to  ^  -'  .) 

bide  .  .  97  99  87  radius  52  75  15  55  45  44 
If  we  require  the  radius  of  a  circle  which  will 
circumscribe  an  octagon  8  inches  on  a  side,  we 
refer  to  column  ?,  take  98  parts  on  the  blade  and 
75  on  tongue,  and  tighten  the  bevel.  As  the  side 
of  a  hexagon  equals  the  radius  of  its  circle,  the 
side  of  an  octagon  must  be  less  than  the  radius; 
hence  we  shift  to  8  inches  that  end  of  the  bevel 


iiniKrVc^f/. 


THE  STEEL  SQUARE  37 

blade  which  gives  the  lesser  number,  in  this  case 
on  t  le  tongue  of  the  square,  as  the  75  parts  to 
which  the  bevel  was  set  are  less  than  the  98. 
The  required  radius  is  then  indicated  on  the 
blade. 

The  following  is  another  table,  to  be  used  for 
the  same  calculations: 

N*'"*^  No.  of  side      Gauge  points 

Triangle 3  1044 

Square 4  8.49 

Pentagon 5  7.06 

Hexagon 6  6.00 

Heptagon 7  5.24 

Octagon 8  4.59 

Nonagon 9  4.05 

Decagon 10  3.71 

Undecagon n  -^^^g 

Dodecagon 12  3.11 

In  a  circle  12  inches  in  diameter,  the  largest 
pentagon  which  may  be  inscribed  is  5.24  inches 
on  a  side.  Hence  for  pentagons  the  bevel  is  set 
at  12  inches  and  5.24  inches.  The  number  oppo- 
site each  polygon  gives  its  side  when  inscribed  in 
a  12-inch  circle. 

The  first  table  is  usually  most  convenient. 

IF/ifji  the  side  of  a  polygon  is  given,  to  find  its 
apothcw  or  perpcndirular.—'^^x.  the  bevel  to  the 
pairs  of  numbers  in  the  table  below.     Thus,  for  a 


r 


38 


PRACTICAL  USES  OF 


heptapron.  set  23  on  tongue  and  25  on  blade,  and 
the  answer  will  appear  on  the  latter. 


Sid 


es 


3    4     5 


8        9       ID      II 


_^  12 

Apothem  9    i     20    13    25    40    40    20    29    28 
Sides    .    31    2    29     15     23     i,2>    29     13     17     15 

The  board  measure.— A  foot  in  board  measure 
is  I  inch  thick  and  i  foot  square.  Set  the  bevel 
to  12  inches  on  the  blade  and  the  length  of 
board  in  feet  on  the  tongue.  Then  move  the 
bevel  to  the  width  of  board  in  inches,  on  the 
blade,  and  the  area  in  square  feet  appears  on 
the  tongue.  Whenever  the  12  inches  is  set, 
whether  on  tongue  or  blade,  there  also  must  be 
set  the  width  of  board  in  inches. 

To  lay  off  degrees  with  the  steel  square,  con- 
sult a  table  of  tangents,  and  from  this  table  take 
the  tangent  of  the  angle  required,  using  the  first 
three  figures  from  the  left  and  calling  them  so 
many  64ths  on  an  inch.     Reduce  them  to  inches, 
and  then,  with  this  quantity  on  one  side  of  the 
square  and  15^  inches  on  the  other  side,  we  v/ill 
have  the  figures  for  laying  off  the  angle.     Tables 
of  natural  tangents  are  usually  calculated  to  the 
radius  unit,  and  are  therefore  decimal  fractions. 
This  method  is  simply  to  multiply  each  by  1,000, 
thereby  obtaining  whole   numbers.     For  exam- 


THE  STEEL  SQUARE  35 

pie.  let  it  be  required  to  lay  off  an  angle  of  10° 
the  natural  tangent  of  which  is  0.176327.    Multi- 
plying this  by  rooo  makes   176.327.     Discarding 
the  de-imal  we  have  176,  and  calhng  the  figures 
64ths  01  an  inch,  we  have  \\\  or  23^  inches.    The 
radius  I   treated  in  like  manner  makes  ^F  or 
^S%   inches.      Now,  taking   2V,    inches  on   'the 
tongue  and  155^  inches  on  the  blade  of  the  square 
the  blade  gives  the  angle  of  10°.  and  consequent-' 
ly  the  tongue  gives  90°  less  10°,  or  80".     There 
are  other  methods  of  laying  off  degrees  with  the 
square,  several  of  which  I  have  described  and 
will  describe  hereafter. 

PROrORTIONAL  REDUCTION  OF  MOLDINGS  OR  OTHER 

WORK 

There  are  many  methods  of  doing    'lis  work 
by  lines,  ordinates.  and  the  pantagraph,  but  I  do 
not  know  of  many  by  the  steel  square.     The  fol- 
lowing, which  may  be  new  to  many  readers,  has 
been  in  use  for  a  long  time:  First  draw  the  mold- 
ing bracket  or  other  work,  as  shown  at  Fig  12  in 
a  square  as  at  A.  B,  E,  F.     Square  out  from'  A 
and  F  lines  meeting  at  B.     Draw  BE,  and  from 
E   measure  off   the   required    projection  of  the 
reduced   bracket,   thus    obtaining   the    point   D 
bquare  down  from  D  to  the  line  BE,  thus  loca- 


:iLxiiyMi^. 


40 


PRACTICAL  USES  OP 


tt 

i 

mm 


ting  the  point  C.     Then  the  line  ED  will  be  the 
width  of  the  reduced  bracket  and  DC  its  hei,;ht. 


Fig.  12 


Now,  at  convenient  points,  their  location  and 
number  being  determined  by  the  nature  of  the 
profile,  as  2,  3,  4,  5,  etc.,  square  lines  to  the  back 
edge,  and  also  to  the  upper  end  of  the  brackets, 
all  as  shown  in  the  sketch.  Take  the  width  ED 
of  the  reduced  bracket  on  the  blade  'of  the 
square,  and  placing  it,  as  shown  in  the  engra- 


jcift-i^-.J   »» 


>'tW"«. 


THE  STEEL  SQUARE 


41 


ving,  against  the  corner  E,  carrying  the  tongue  of 
square  up  until  its  edge  strikes  the  outer  corner 
A  of  the  original  bracket,  draw  a  line  along  the 
blade,  all  as  indicated  by  KE.      Square   down 
from  points  in  this  line  to  the  points  in  the  upper 
Hie  of  bracket,  i,  3,  4,  5,  etc.,  already  obtained. 
Take  the  length  of  the  line  DC  on  the  square, 
which  is  the  length  of  the  bracket  after  reduc- 
tion, and  place  it,  as  shown  by  the  shaded  square 
in  the  sketch,  at  E.     Carry  the  square  up  until 
the  blade  strikes  the  corner  at  F.     Mark  along 
the  blade  of  the  square,  thus  producing  the  line 
EH,  which  is  the  back  of  the  diminished  bracket. 
From  this  line  square  out  the  line   EG  indefi- 
nitely;   also  square  out  the  lines  16,   15,   14,   13, 
etc.,  from   the  points  in  the  back  edge  of  the 
original    bracket,   extending    them    indefinitely 
across  the  space  the  reduced  bracket  is  to  occupy. 
Tal     a  straight-edge  and  place  it  against  the  line 
KE  and  mark  the  points  that  have  already  been 
obtained  in  it.     Then  transfer  these  distances  on 
to  the  line  EG.     If  preferred,  this  may  be  done 
by   the   compasses,   setting   one   leg   at    E    and 
describing  arcs  from  the  several  points  in   KE, 
striking  the   line    EG.      From    the    points   thus 
located  in  EG  lines  are  then   to  be  carried  at 
right  angles  to  it,  being  produced  until  they  meet 


.    1 


iw^'«».:«%v  "^^w 


4» 


PRACTICAL  USES  OP 


he  lines  drawn  from  corresponding  numbers  on 
he  inner  edge  of  the  original  bracket.     Then  a 
line  traced  through  these  intersections  will  pre 
duce  the  profile  of  the  bracket  diminished.     The 
number  of  f\xed  points  in  the  profile  of  the  orig- 
inal bracket  necessary'  to  be  used  will   vary  in 
different  cases.     Where  the  lines  are  long  and 
regular  less  will  be  r^.quired  than  where  they  are 
short  and  irregular.     To  increase  the  size  of  a 
g.ven  bracket  the  process  here  described  is  to  be 
reversed.     The  same  general  rule  may  be  also 
applied  m  drawing  the  profile  of  raking  mold- 
jnp      I  thmk  it  will  be  seen  that  I  have  not  here 
laid  down  an  arbitrary  rule.     The  principle  on 
which  It  is  founded  is  in  laying  down  a  line  the 
ength  of  the  required  bracket,  and  dividing  that 
line   m   the  same    proportion   as    the    original 
bracket. 

//  zs  required  to  get  the  length  of  a  hoop  for  a 
wooden  tank,  by  the  steel  sguare.-^o  accomplish 
th.s.pioceed  as  follows:  First  produce  a  circle 
to  any  desired  scale,  say  i  inch  to  the  foot,  and 
this  on  24  feet  would  be  24  inches.  Then  place 
the  heel  exactly  at  the  center  as  indicated  in 
F«g.  13.  and  scribe  closely  to  the  square,  cutting 
the  circumference  of  the  circle,  as  indicated  by 
BC.      I  hen    draw    the    chord   intersecting  the 


'4 


THE  STEEL  SQUARE  ^3 

points  n.C  already  referred  to.  The  next  step  is 
to  divide  the  segment  equally,  which  is  done  by 
the  line  DE.  Now 
three  times  the  di- 
ameter plus  the 
distance  DE  will 
give  the  required 
measure  men  .  or 

circumference. 
To  perform  this 

by    figures   alone, 

take  the  diameter 

of  the    tank    and 

multiply  it  by 


Fig.  13 


31416.     If  the  diameter  of  the  tank  is  24  Teet.  for 
-amp  e   the  equation  is  as  follows:    .4x3.1416 
/5.09.S4,    or  multiply  the  diameter  by  22  and 
divide  by;.     Thus  24x22-^7  =  75^ 


Fig.  14 
The  prober  angle  for  ordinary  door  and 


win- 


. BEfi 


! 

J 


44 


PRACTICAL  USES  OP 


dow  s.lls  is  about  ,  inch  drop  to  the  foot.  A 
motho<i  of  finding  this  indination  c,uickly  is 
shown  in  Fig.  ,4.  The  square  n.ay  I..  lai<i  on 
the  rdirr  of  th,-  bench,  or  on  the  rd^r  of  a 
boar<l.  and  the  bevel  set  to  suit  in  short  order. 

SOME    MISrF.LI.ANKOrs    I'ROKLKMS 

These  problems  are  gathered  chiefly  from  the 

Saent.f.c  American"  Supplement,  and  were  orig- 

•nally  contributed  by  Mr.  O'Conneli.     Some  of 

them  are  more  curious  than  useful,  but  all  are 

inten^sting. 

The  arms  of  a  straight  horizontal  lever  are  8 
and  ,2.  A  weight  of  9  lbs.  is  suspended  from 
the  shorter  arm;  what  weight  will  balance  it  on 
the  longer  arm?  Set  to  i.  on  blade  and  Son 
arms.  Move  the  bend  to  9  on  blade,  and  6  is  the 
answer  on  tongue. 

What  power  is  required  to  support  a  weight  of 
4  lbs.  on  an  incline  of  5  in  30?  Set  ,  on  tongue 
and  .30  on  blad..  Move  the  bevel  to  4  (lbs.)  on 
blade,  and  -',  (lb.)  is  obtained  on  tongue. 

A  body  is  weighed  in  a  false  balance  and  in 
one  scale  appears  to  be  0  o..  and  in  the  other 
12  oz.  Wha:  ,s  its  true  weight?  I-ind  a  mean 
proportional  between  these  numbers,  that  is  the 
square     root     of     their     product.       gxi^Jios' 


THE  STEEr,  KyUAKE 


..o«-,o.,,cA„s.    Th.  ,an,o  e«„„.,e  is  solved 
b        cs„,y,ai<i„,,,o.;,Mf,hesu„,of, 

•'"     '-  ;'"''  "^"  "'"'^h  i»  ">-  .lifr.T,.ncc  between 
10 !i  and  9,  or  I'     Thi.  im    •  ,        "•""'-" 

nuse  nf  ■    .*- '°  •'»"""•''<■  hypothe- 

a        ,h     ■'"    '""■""•■""*■    ^'«'"-«l"l    triangle, 
^'-l   ■he    ,>,   one  si,!,-.      I,    has    already  been 

:::;::*=" ""- '°  '■-"  •■-  -'-  s-i.-,  or ;: 

.|.-         ;^-'""™«  square:  what  is  the  diam. 
;>l.ndn,:al  one  with   ,h,.  same  area  of 

,    u      ::,^'^'-     "— - -n  <"n«„e  and  we 
'iiitaiii  22,  th<-  answer  .,n  blade 

Tl,e  length  and  an«l,:s  of  a  brace  of  irregular 
"■n.  ".•  any  r„n,  ,nay  be  fonn.l  rea.lily  by  at  K, 
^nsr  th<;  following  ml,,   usin.r  ,1,  "  ^ 

- ^•''"".n;,,,ad:To^tie"'Takrr 

-™nle,  the  ease  o,  a  brace  of  whi,h    ht'un   : 
.^"  '""^t  -''  •"-=  l-i«lu  37  inches.     Me.,     in 
.»  ross  the  space  for  the  length  of  the  brae    ""h 
^    square  will  not  do;   accortlinRly  then  re 
•  ";  '"o  lengths  by  four,  which'^i';  :,";;; 

arms  of  the  sqnare  and  measuring,  across 
','  --l-.-J  l---ng.h  will  be  obtained.     To  do  this 
take  a  p.ece  of  board,  join,  one  edtre  an,   r 
iraufrp  rmri-   f  i  ^^'^t-  and  run  a 

feduye  mark   from   the  erJtrp  fKo    i     •      . 

eage  the  desired  width. 


46 


PRACTICAL  USES  OP 


j 
i 


i 


II 


Then,  placing  the  square  so  that  the  figures  fall 
on  the  gauge  mark,  apply  it  four  times,  scribin.r 
along  the  blade  and  tongue  respectively  for  the 
two  ends.     This  gives  the  net  length  of  the  brace 
and  the  proper  cuts  for  the  joints.     This  may  not 
be  the  best  rule  which  can  be  employed  for  the 
purpose,  but  it  is  short  and  simple.     Any  ordi- 
nary carpenter  can  work  it,  and  it  is  undoubtedly 
correct. 

At  this  point  I  show  a  few  quick  rules  to  give 
a  square  stick  an  octagonal  shape.  The  rules 
given  on  the  side  of  the  square  as  shown  in  my 
description  of  squares  in  the  first  volume,  while 
being  perfectly  correct  so  far  as  they  go,  do  not 
work  so  well   where  fractions   of   an    inch    are 

involved,   so    the 
following  methods 
of   finding   the 
points   for  the 
gauge   lines    are 
shown  at  Pig     jc 
which  shows  the  square  as  laid  obliquely  across 
the  face  of  the  stick  so  that  just  6  inches  will  be  on 
the  stick.     At  iH  inches  from  each  corner  draw 
gauge  Imes.  which  will  be  the  correct  corners  for 
the  octagon.     If  the  timb.-r  is  over  six  inches 
square  lay  twelve  inches  of  the  square  upon  the 


''IG.    15 


.JJ^^f^^^ 


*1L-^-, -J^ 


W^- 


THE  STEEL  SQUARE  ^y 

face  and  gauge  at  3^  inch  -s  from  the  corner     If 
over  12  inches  wide,  layover  me  whole  24  inch^ 
of  the  square  and  prick  off  7  inches  from  each 
end.  and  these  points  will  be  the  gauge  points. 
Indeed,  .t  is  best  to  use  the  whole  length  of  the 


Fig.  16 

sqtiare  in  laying  off  work  of  this  kind,  and  prick- 
•n^^  off  7  inches  from  each  end  of  the  square   nn 
-tter  what   may  be  the  width   of  the  tX 
P^g.  16  shows  this  method  quite 

^-'early.     The  <lotted  lines  rep- 

resent  the  gauge  marks.     Fig. 

17   shows   a   section    through 

one  corner  of  a  timber;  B  and 

^  '•^^Present  the  gauge  marks 

on  an  adjacent  face  of  the  tim-  F,...; 

ber.  and  are  oath   -  .-^  u 

^^e  each   ,   mches  removed  from  the 


48 


PRACTICAL  USES  OP 


I 


1 


I     ■ 


corner  A.  The  timber,  of  course,  is  to  be  dressed 
until  the  Hne  BC  becomes  a  surface.  Now,  if  the- 
rule  were  accurate,  the  diagonal  BC  in  Fig.  17 
should  measure  exactly  10  inches,  because  the 
adjacent  faces,  by  the  rule,  would  be  laid  off  by 
measurement  to  that  dimension,  aiA  all  the  faces 
of  the  octagon  should  be  the  same  dimension. 
That  BC  in  this  case  is  not  exactly  10  inches  may 
be  easily  proven.  Since  CAB  is  a  right  angle, 
the  length  of  CB  must  be  equal  to  the  square  root 
of  the  sum  of  the  squares  of  BA  and  AC  {7x7- 
49,  49+49=9^).  the  square  root  of  which  is  less 

E  than  10,  the  square 
1  of  lobeifig  100.  It 
*  follows  that  a  stick 
of  timber  reduced 
to  octagon  shape 
by  this  rule  will 
have  four  of  its 
sides  ic  inches  in 
width  and  four  of  its  sides  a  fraction  less  than  10 
inches,  equal  to  the  square  root  of  98.  Fig.  18 
shows  the  same  thing  in  a  little  different  shape, 
by  representing  a  partial  section  through  a  tim- 
ber 24  inches  square.  The  gauge  marks  C  and 
D  are  each  7  inches  from  the  corners  A  and  B; 
the  side  CD  is  10  inches,  but  the  sides  CB  and 


Fig.  18 


i 


THE  STEEL  SQUARE 


49 


DF  are  less  than  lo  inches.  As  this  rule,  like 
some  others  in  common  use,  is  supposed  to  be 
mathematically  accurate,  I  think  the  diagrams 
introduced  will  be  of  interest  to  readers. 


i:ll 


n 


SOME  GOOD  THINGS 

A  whole  bunch  of  good  things,  some  original, 
some  gathered  from  the  four  quarters  of  the 
earth,  are  shown  in  the  two  illustrations  follow- 
ing, which  have  been  handed  me  by  Mr.  Stod- 
dard, to  include  in  my  "Odd  Things  Done  by  the 
Steel  Square."  Some  of  thee  examples  have 
already  been  given  under  various  heads,  but  a 
repetition  is  excusable  when  placed  before  the 
reader  in  another  light.  These  problems  were 
published  in  "The  Carpenter"  some  time  ago, 
and  were  well  spoken  of  by  a  large  number  of 
woodworkers. 

After  stating  that  he  was  much  indebted  to 
Hodgson  for  his  knowledge  of  the  steel  square, 
Mr.  Stoddard  goes  on  to  say  to  the  workman 
"that,  to  advance  rapidly  he  should  be  a  faithful 
student  and  observing,  and  should  notice  how 
every  new  piece  of  work  is  done,  and  should  pur- 
chase some  good  works  on  the  subject  in  hand." 
Then  he  goes  on  to  describe  the  various  meth- 


so 


PRACTICAL  USES  OF 


ods  of  roof  framing  as  illustrated  in  the  seven 
illustrations  shown  in  Fig.  iq,  as  follows: 


■P^ 


BY  O.L3T;oo/^f\a.     | 


With 
4;  <STE:E:L(6QV//\l^b 


Rsor  t(«14 

VJPTCM 


l^«n/,  i^yw 


r:^    w.-     ■■■■--    -,■  ■    \- ■  -     ■-■■.''■■-'■■''■-.  ■:'^- 
"iT^i.-^NV  J, -;•>;;,  '.-a;^  j,.;;v.  t..:v-;  /.-?x  »  ■: 
V-'    vr;.-'  \v^.'.'  \'<;-'   >\:-''  v,-;,-'  'ov;;-'  '%,» 

Hif» 


■.> 


l^(>if'-«  Rin    ,-\  ->  ,.A  ,^  ,-\  ,-•»  ^a 

^    '■  ,^>'.  '       J  •'•-', '^^-r^ ,■■»!■  j'^ii  pf..ii  J    r.  II         ■Jli'gA 

\>    ^»  V  O  -J  u  O  0  >^ 


>kC«/V^ 


B<</'(u;'* 


\^9T>t,nihi 


l£r/»TV^(^ijt 


Fig.  19 


ROOF   FRAMING 

In  th*^,  cut  I  have  illustrated  a  %  pitch  hip  roof, 
16x24  feet,  rafters  16  inches  apart. 

MAIN    RAFTER 

One-third  pitch  rises  8  inches  to  the  foot,  and 
as  8  feet  is  half  the  width  of  the  building,  the 
run  must  be  8  feet.  Therefore  put  the  square  on 
12  and  8  and  eight  times  gives  the  length  and 
bevels  (as  illustrated).  Notice  how  it  is  squared 
up  at  heel,  and  amount  allowed  for  ridge. 


1 


THE  STEEL  SQUARE 


51 


CENTER    nil'     RAFTER 

As  the  diagonal  of  a  foot  is  17  inches,  take  17 
inches  on  blade  in  place  of  12,  and  we  have  hip 
rafter  (as  illustrated). 

Now  these  methods  are  not  new  or  original,  as 
they  have  probably  been  used  for  ages,  yet  it  is 
surprising  how  few  carpenters  know  them. 

JACK    RAFTERS 

My  method  for  jacks  is  an  original  idea  to  me, 
yet  it  may  have  been  used  before  1  was  born. 

I  simply  lay  the  square  on  the  same  as  for  com- 
mon rafter.  If  you  wish  them  16  inches  apart, 
move  th^  square  up  to  16  inches;  if  18  inches 
apart,  move  up  to  18,  etc.  The  side  cut  is  the 
length  and  run.  Cut  on  length.  If  you  wish  to 
bevel  top  of  hip,  take  length  and  rise.  Cut  on 
rise. 

OBSERVE  ALL  THE  ILLUSTRATIONS 

Now  remember  the  same  method  applies  to  all 
pitches.  Run  the  same;  simply  change  the  rise 
to  whatever  rise  the  roof  is  to  the  foot.  This 
applies  to  cornice  as  well  as  rafters. 

Do  not  be  satisfied  with  this  knowledge,  but 
study  the  use  of  the  square  and  go  further,  as 
there  is  no  'imit  to  what  can  be  accomplished 
with  it. 


r 


5» 


PRACTICAL  USES  OF 


i 


PRACTICAL   USE    OF   SQUARE   AND  RULE 

Study  and  fully  understand  the  eight  illustra- 
tions in  this  one  little  cut,  and  you  will  find, 
by  thought  and  application,  as  the  occasion 
requires,  you  have  learned  a  great  deal,  as  you 
will  readily  learn  more. 

If  you  have  a  board  7  inches  wide  and  wish  to 
divide  it  into  four  equal  parts,  turn  the  rule  until 


Fig.  20 

it  strikes  eight  inches,  and  mark  at  each  2  inches, 
as  in  No.  i.  Fig.  20. 

I  use  that  almost  daily  not  only  in  ripping  up 
hoards  but  in  drawing,  etc 

If  you  happen  to  wish  to  square  a  board  and 


THE  STEEL  SQUARE 


53 


do  not  have  a  square,  take  a  rule  and  apply  as 
shown  in  No.  2,  Fig.  20. 


IN   LAYING   OFF    RAFTERS 

Some  may  not  like  to  place  the  square  on  once 
for  every  foot  of  run,  as  I  illustrated  in  another 
cut.  Also,  if  it  is  to  go  to  a  given  height  one  may 
not  wish  to  stop  to  figure  the  exact  rise  to  the 
foot,  figuring  out  the  fraction,  etc.  Take  a  roof 
to  be  7  feet  3  inches  high  and  run  8  feet  5  inches. 
Put  your  rule  on  71?  and  SA,  and  you  will  have 
VV  or  1 1  feet  2  inches  length  of  rafter,  as  seen 
at  No.  3,  Fig.  20. 

If  you  wish  to  hip  the  same  roof,  as  it  is  8  feet 
5  inc  ^is  to  the  deck,  the  run  of  hip  must  be  the 
diagonal  of  8  feet  5  inches,  which  is  11  feet  11 
inches,  as  illustration  4.  The  run  being  11,  11, 
and  the  rise  7,  3,  place  the  rule  on  them,  and  we 
have  14  feet,  as  shown  in  No.  3,  Fig.  20.  If  you 
are  buying  lumber  at  $13  per  M,  and  you  wish 
to  know  what  7000  feet  costs,  place  the  square 
on  10  and  13,  bring  it  down  to  7  on  the  tongue, 
and  we  will  find  we  have  qtV  on  blade,  or  $9,10, 
which  is  the  correct  answer,  as  shown  in  No.  6, 
Fig.  20. 

If  you  wish  to  strike  a  circle  and  have  nothing 
but  a  rule,  apply  as  shown  at  No.  7. 


n  i-: 


11 


PRACTICAL  USES  OP 


One  noon  a  large  crowd  of  workmen  was 
asked  by  the  foreman  how  to  cut  a  third-pitch 
rafter  so  it  would  lay  on  half-pitch  roof.  It 
seemed  to  me  to  lay  off  half-pitch  and  then  from 
that  half-pitch  line  lay  off  %  would  cut  it.  I 
tried  it,  and  we  were  all  surprised  to  find  it 
O.K.,  as  shown  at  No.  8,  Fig.  20.  All  of 
these  problems,  as  formulated  by  Mr.  Stoddard, 
are  valuable  in  themselves  because  of  their  sim- 
plicity and  because  of  their  paving  the  way  to 
many  other  things.  Indeed,  as  I  have  often 
stated,  there  appears  to  be  no  limit  to  the  use  of 
the  square;  and  I  am  sure  there  are  hundreds  of 
workmen  scattered  over  the  country  that  have 
found  out  things  that  can  be  done  with  this  tool, 
of  which  we  never  hear,  and  I  would  like  to 
impress  on  the  minus  of  the  readers  of  these 
volumes  the  fact  that  if  they  have  any  new 
"kink"  they  have  worked  out  with  the  square, 
they  will  be  doing  a  public  good  by  sending  a 
description  of  same  to  the  publishers  of  this 
work,  so  that  it  may  be  published  in  future  edi- 
tions, and  thus  saved  to  the  trade. 

TO  OBTAIN  THE  LENGTH  OF  A  HOOP  FOR  A  BARREL 
OR  TANK  BY  THE  STEEL  SQUARE 

There   are  a  number  of  ways  by  which  the 
length  of  a  tank  or  barrel  hoop  may  be  obtained, 


fe 


THE  STEEL  SQUARE 


55 


some  of  them  being  much  easier  than  the  one  I 
am  about  to  describe,  as  using  a  traveler,  for 
instance,  after  the  tub  is  standing,  or  stretching  a 
tape-line  arour.J  the  tub,  and  other  ways;  but 
where  these  methods  cannot  be  applied — whiclr 
is  very  often — then  the  following  method  may 
be  employed  with  profit:  The  diameter  being 
known,  the  circumference  may  be  obtained  by 
the  ordinary  rule  of  multiplying  the  diameter  by 
3.1416,  which  will  give  the  circumference  nearly, 
then  take  a  pair  of  dividers  and  strike  a  circle  to 
a  scale  of  say  >^  or  K  inch  to  the  foot;    then 


pl  ace  the  outside  corner  of  a  steel  square  to  the 
center  of  the  circle,  as  at  A,  Fig.  21.  Referring 
now  to  the  sketch,  scribe  along  the  outside  of  the 
square  from  B  to  A,  and  from  A  to  C,  then  draw 


56 


PRACTICAL  USES  OF 


I 


a  line  from  B  to  C,  intersecting  at  the  points 
where  the  lines  previously  drawn  cut  the  circum- 
ference of  the  circle.  Now  obtain  the  center  on 
the  line  BC,  as  at  D.  Take  the  square  and  place 
it  with  one  edge  at  the  center  of  the  circle,  cut- 
ting the  line  BC  at  D,  and  draw  the  line  DE. 
Multiply  the  diameter  of  the  circle  by  3  and  add 
the  distance  from  D  to  E.  For  example,  sup- 
pose the  tank  is  24  feet  in  diameter;  the  circum- 
ference would  be  75  feet  6  inches;  thus  3x24=72 
plus  the  distance  from  D  to  E,  which  is  3  feet  6 
inches,  making  75  feet  6  inches.  The  sketch 
so  clearly  shows  the  method  that  further  expla- 
nation would  appear  to  be  unnecessary. 

TO   MEASURE    INACCESSIBLE    DISTANCES    BY   THE    A'D 
OF    THE    SQUARE 

A  number  of  wTiters  on  the  s', -1  square  have 
written  on  this  subject  and  ha\  rot  the  matter 
down  fine;  but  the  best  of  the  a.  .icles  I  have  met 
with  are  those  of  Lucius  Gould  of  Newark,  N.  J., 
and  A.  W.  Woods  of  Lincoln,  Neb.,  the  latter, 
in  my  opinion,  being  the  better  of  the  two,  and  it 
is  from  the  latter  that  the  following  is  largely 
taken,  as  it  is  placed  before  the  readers  in  a  sim- 
p     unostentatious  manner. 

Every  mechanic  knows  that  a  triangle  whose 


THE  STEEL  SQUARE  57 

sides  measure  6,  8  and  10  forms  a  true  right 
angle  and  is  the  method  commonly  used  in  squar- 
ing foundations.  But  how  many  ever  stopped 
to  think  what  other  figures  on  the  square  will 
give  the  same  result? 

By  referring  to  trigonometry  we  find  only 
three  places,  using  12  inches  on  the  tongue  as  a 
basis  and  measuring  to  the  inches  on  the  blade 
that  do  not  end  in  fractions  of  i  inch  on  the 
hypothenuse  side.  They  are  as  follows:  12  to  5 
equals  13,  12  to  9  equals  15,  and  12  to  16  equals  20. 
Now,  as  we  usually  use  a  lo-foot  pole  to 
square  up  a  foundation,  we  find 
that  all  of  the  above  contain  lengths 

greater  than  our  pole,  so  we  must 

take  their  proportions.     The  first 

contains   numbers  not  divisible 

without  fractions,  consequently  we 

will  pass  on  to  the  next.     We  find 

that  three  is  the  only  number  that 

will  equally  divide 

all   the    numbers 

with  quotients,  as 

follows:  4,  3,  and 

5,  but    these    are 

too  srncul  to  obtain 

t  h  ('   best  results.  fig.  22 


5« 


r.M 


i  I 


PRACTICAL  USES  OF 


Now  let  us  examine  12.  16.  and  20.     They  are 
«'vpr>  numbers,  and  are  divisible  by  2  and  4.  Fig 
22       ^f  we  take  one-half  their  dimensions,  we 
have  *:,  8.  and  10. 

"he  .    beinjr   convenient    length^   and    easily 
rcmrmoered.  custom  has  settle  1  on  tliese  figures, 
i  h:   .  are  other  places  that  6,  «.  an  «  lo  can  be 
''snd  t  )  ad. an    .;^e. 

Simpi..;e  /or  .omc  reason  we  want  to  know  the 
C)stan<o  pcross  a  body  of  water.  We  cannot 
v^ade  .!.  iMithcr  can  we  depend  on  a  line 
stretched  across,  because  when  it  is  re-stretched 
on  an  accessible  place  of  measurement  we  have 
no  way  of  determining  when  it  is  drawn  to  the 


Fir,.    23 

same   tension.      Now,  referring  to   Fig.  23,  we 
wrint  to  find  the  distance  from  A  to  H.     Lay  off 
the  angle  of  6.  8.  and   i^.  at  hot'-    A    -r  l  V 
shown. 


THE  STEEL  SQUARE  5, 

Sinco  the  base  and  perpendicular  of  a  n^ht 
anj^Hed  triangle  are  of  equal  lengths  when  the 
hypothtnuse  rests  at  45'  with  the  former,  we 
measure  off  6  feet  on  the  8-foot  side  as  shown, 
and  this  will  be  the  point  of  sight  from  A.  With 
a  man  sicfhtin^j  from  both  A  and  M,  a  third  sets 
the  stake  at  C.  Then  BC  must  be  the  same 
length  as  AB.  (The  arc  i^  shown  here  to  prove 
the  accuracy  of  rhe  diagram.) 

By  measuring  from  A  -nd  B  to  he  water's 
edge  and  sui)trac.  ng  th-  amount  fro.u  BC  will 
be  given  the  width  of  the      )dy  of  water. 


Frr..    _4 

^ig.  -  dlustiates  how  1  ree  or  inaccessible 
hcii^ht  can  be  measrr<  d  on  the  same  principle 
wuii  the  aid  of  the  st  ei  ^quarf  Take  a  straight- 
tdg.   an  i  fasten  at  any  of  the  equal  figures  on 


6o 


PRACTICAL  USES  OP 


the  tongxie  and  blade.     Level  and  set  as  shown, 
and  the  base  will  be  equal  to  the  perpendicular. 

MAKING   TRESTLES    BY   THE   AID   OF  THE    STEEL 

SQUARE 

The  usual  batter  given  to  trestles  is  3  inches  to 

the  foot,  but  any  figures  may  be  taken,  according 

A^^  to   the  amount  of  batter 

:«'  "  I  bi  M  I  T-rfr-r-T-r-ri  required. 

■_       I  7  The   manner  of  using 

the  square  to  obtain  the 
proper  angles  and  bevels 
is  shown  at  Fig.  25,  which 
shows  how  the  vertical 
and  horizontal  cuts  can  be 
determined. 

Fig.  26  shows  the  end 
of  a  framed  trestle,  with 
posts  on  the  same  inclina- 
tion as  shown  at  Fig.  25. 
It  also  shows  braces,  one  framed  in  the  angles 
and  the  other  spiked  on. 

This  problem  may  be  ap- 
plied to  all  tapering  frame- 
work when  the  taper  or  lean        /  /  x 

is  in  one  dire  ct  i  o  n  only,    r'y-y-- —— 1 

V;hen  the  work  is  pyrami-  pzo  26 


Fig.  25 


THE  STEEL  SQUARE 


6i 


dal,  or  leaning  two  ways  at  the  corners,  then  a 
different  mode  of  obtaining  the  bevels  must  be 
adopted.    I  will  refer  to  this  again. 

SOMETHING  MORE  CONCERNING  ROOF  FRAMING 

In  cutting  the  timbers  for  a  hip  or  valley  roof, 
or  for  both,  it  will  readily  be  seen  that  the  prin- 
ciple involved  in  getting  the  lengths  and  bevels 
is  the  same  as  for  getting  the  lengths  and  bevels 
for  braces,  for  if  you  compare  the  top  end  of  a 
brace  with  the  ridge  and  valley  rafter  in  a  roof, 
it  will  soon  be  seen  that  they  are  the  same. 

L-t  us  take  a  square  stick,  say  a  4x4.  :c  will 
form  a  half-pitrh.  If  the  brace  is  set  a^  an  angle 
of  45'.  you  get  the  bevel  as  shown  at  Fig.  27. 


Now  take  one-half  of  the  diagonal  width,  as  at 
Ali,  and  measure  this  distance  along  the  edge  of 
the  stick,  as  at  CD.  Square  over  to  the  other 
-ngle  and  join  DF.  Measure  as  before,  making 
EF  equal  to  CD.  Square  across  timber  again, 
and  FG  will  be  the  cut. 


6a 


PRACTICAL  USES  OP 


For  a  simpler  way,  that  will  suit  all  braces  and 
at  any  angle,  take  a  piece  of  timber  and  mark  as 
above.  Suppose  you  have  a  4x4  to  be  cut,  say  6 
feet  from  a  post,  and  run  up  on  the  post  8  feet  6 


inches.     Lay   he  square  on  the  lines  laid  down 
thus  (see  Fig.  28J: 
Transfer  the  distance  AB  in  Fig.  28  to  the  side 

ft      B C 2_ 


¥*¥ 


%■ 


n;.' 


Fig.  29 

of  the  timber  in  Fig.  29.  Square  over  to  the 
other  edge,  and  join  the  angles  as  before;  then 
cut. 

Another  way:  Take  your  square  and  find  the 
distance  across  from  8>^  on  one  arm  of  the 
square  to  6,  Fig.  30,  on  the  other  same  as  for  a 


THE  STEEL  SQUARE 


63 


common  rafter.  This  distance  will  mark  the 
side,  using  the  line  formed  by  the  arm  of  the 
square  with  the  6. 


Fig.  30 

Cross  over  on  the  other  side  and  cut,  as  shown. 


Fig.  31 

AB,  Fig.  31,  represents  the  ridge  or  edge  of 
the  4x4;  line  to  the  left  of  B  the  valley  or  side  of 
the  4x4.  It  will  be  noticed  that  the  ridge  A  is 
one-half  the  width  of  the  width  B. 

Of  course  this  proposition  holds  good,  and 
practical  experience  has  so  proven  it.  Any  dis- 
tance may  be  used  instead  of  those  given,  by 
applying  the  instructions  as  shown. 


lii     ! 


1" 


1 


3f  t^t^ 


^-«ST 


64 


PRACTICAL  USES  OP 


These  instructions  apply  more  particularly  to 
roots  when  the  pitches  are  equal,  but  there  are 
many  cases  where  the  pitches  are  not  the  same 
on  each  side  of  the  roof,  and  to  meet  this  in- 
equality the  following  diagrams  and  explanations 


Fig.  3^ 

are  given:  Let  us  examine  Fig.  32,  here  we  have 
the  valley  AB  and  the  run  of  the  common  rafters 
AC  and  AD,  of  unequal  lengths.  To  obtain  the 
cuts  for  the  top  of  the  valley  rafter  draw  AE  and 


I 

M 


THE  STEEL  SQUARE 


05 


AF  at  right  angles  to  AB,  extending  them  to  the 
ridge  line.  Now  AI^  and  the  length  of  the  rafter 
on  the  square  gives  ihe  cut  ABE,  and  AF  with 
the  rafter  gives  the  cut  FBA,  marking  on  the 
side  representing  the  rafter.  The  lengths  of 
these  auxiliary  lines  may  be  obtained  by  laying 
the  square  on  the  roof  plan  and  noting  their 
lengths  by  the  scale  of  the  drawing,  or,  better 
yet,  by  a  little  simple  proportion,  as  exemplified 
in  Fig.  33,  of  the  sketches— AC  :  AD  ::  AB  :  AE. 


Fig.  33 

That  is,  take  the  runs  of  the  common  rafters, 
as  AC  and  AD,  on  the  square,  place  them  on  the 
edge  of  a  board  and  mark  along  AC;  then  slide 
the  square  on  the  line  AC  until  A'C  equals  CD; 
then  A'E  equals  AE  of  Fig.  32.  For  the  valley 
jacks  use  the  length  of  rafter  over  AD  with  run 


66 


PRACTICAL   USES  OF 


1    I 


AC  for  the  angle  DAB  and  the  reverse  of  the 
opposite  side.  It  is  seldom  if  ever  that  a  draw- 
ing  is  necessary. 

On  the  subject  of  the  steel  square  as  used  in 
laying  out  roofs.  I  cull  the  following  from  an 
English  source,  which,  while  containing  nothing 
new  to  those  who  have  made  a  study  of  the  steel 
square  and  its  applications,  yet  is  interesting  as  it 
offers  another  side  light,  as  it  were,  to  the  sub- 
ject, and  contains  some  things  that  may  prove 
mstructive. 

Regarding  the  lengths  and  cuts  of  hip  rafters 
on  a  pitch  of  45°,  the  principle  is  the  same  in  all 
pitches.    Take  the  run  and  rise  of  the  common 
rafter  on  both  tongue  and  blade,  and  measure 
across,  and  the  length  of  the  common  rafter  is 
ascertained.     Take  the  run  on  both  tongue  an^ 
blade,   and   measure   across,  take   the   distance 
obtained  on  the  blade  and  the  rise  (12  inches)  on 
the  tongue,  and  measure  across  again,  and  the 
length  and  bevels  of  the  hip  rafter  are  found. 
Ihe  above  is  to  i-inch  scale. 

Again,  by  taking  12  on  both  tongue  and  blade 
and  measuring  across,  the  actual  length  of  rafter 
for  I -foot  run  is  found.  Take  17  on  the  blade 
and  12  on  the  tongue,  and  measure  across,  and 
the  length  of  hip  rafter  for  i-foot  run  of  common 


THE  STEEL  SQUARE 


67 


rafters  is  found;  that  is— if,  say,  the  half-width  of 
the  building  is  15  feet,  take  for  the  length  of  the 
hip  the  length  of  i-foot  run  15  times,  which  is 
always  17  on  the  blade,  and  the  rise  for  i  foot  on 
the  tongue.  If  the  instructions  given  are  fol- 
lowed anyone  will  be  able  to  get  all  the  cuts, 
lengths  and  bevels  for  a  roof  of  any  pitch  what- 


ever. 


Fig.  34 

Supplementing  the  foregoing  it  maybe  said 
that  the  line  diagram,  Fig.  34,  shows  the  plan  of 
one  end  of  a  hipped  roof,  the  elevation  of  a  pair 
of  common  rafters,  and  the  development  of  the 
iour  quarters  of  the  hip.  These  will  be  sufficient 
to  show  clearly  how  the  steel  square  is  applied. 


'    1 


■nv-mm^- av^vLa^a  r  .-rm 


-rai 


68 


PRACTICAL  USES  OP 


As  both  sides  of  the  hip  are  alike,  I  have  on  the 
left  side  of  the  hip  developed  only  one  side.    The 
process  is  as  follows:  First  drop  the  point  of  the 
common  rafter  A  to  A',  and  draw  a  line  from  it 
to  corner  B.     If  this  diagram  is  made  on  card- 
board to  a  scale  of  %  inch  to  i  foot,  and  the  tri- 
angle formed  cut  through  with  a  penknife  from 
A'  to  B,  and  from  A'  to  C,  leaving  from  C  to  B 
as  a  hinge,  also  cutting  through  the  lines  from  C 
to  A,  and  from  A  to  E,  and  folding  this  up  on 
the  line  C  to  E  as  a  hinge,  raising  the  other  tri- 
angle up  and  letting  it  rest  on  the  first,  one  side 
of  the  hip  will  be  represented  in  the  position  it 
would  occupy  when  fixed;    the  points  A  and  A' 
would   stand    plumb  over  the  point   D.      Now 
apply  the  steel  square,  and    note  its    position. 
Lay  the  tongue  on  the  line  C  to  B,  which  equals 
the  run,  and  the  blade  on  the  line  C  to  A,  which 
is  the  length  of  the  common  rafter,  while  from  B 
to  A'  is  the  length  of  the  hip  rafter.     Marking 
alongside  the  blade,  it  will  be  seen,  must  always 
give  the  bevel  for  jack  rafters.    The  numbered 
lines  represent  the  jack  rafters. 

On  tne  right  side  of  the  hip  both  carters  are 
developed,  because  the  run  of  one  is  :>  feet,  and 
t'.at  of  the  other  14  feet.  Now  note  the  differ- 
ence in  the  application  of  the  square.    Take  the 


THE  STEEL  SQUARE 


69 


run  of  the  common  rafter  on  the  end,  on  the 
tongue,  that  is,  from  E  to  F  (not  from  E  to  D), 
and  the  length  of  the  common  rafter  on  the  right 
on  the  blade,  and  mark  by  the  blade.  This 
gives  the  top  cut  or  bevel  for  the  jack  rafters  on 
the  right.  Now  take  the  run  of  the  common 
rafter  on  the  right  side — that  is,  from  D  to  E  on 
the  tongue,  and  the  length  of  the  common  rafter 
on  the  end  (which  is  the  same  length  as  on  the 
left  side)  on  the  blade,  and  mark  by  the  blade. 
This  gives  the  top  cut  of  the  jack  rafters  for  the 
end.  If  the  triangles  are  cut  and  placed  in  posi- 
tions as  suggested  for  the  other  side,  the  correct- 
ness of  the  measurement  will  be  demonstrated. 
The  length  and  bevels  of  the  hip  on  the  right 
side  are  obtained  by  taking  the  run  of  the  end 
(16  feet)  on  the  blade,  and  the  run  of  the  right 
side  (14  feet)  on  the  tongue,  and  measuring 
across,  then  taking  the  length  thus  obtained 
(which  is  the  run  of  the  hip  rafter)  on  the  blade 
and  rhe  rise  (10  feet  8  inches)  on  the  tongue,  and 
measuring  across,  which  giv^s  the  length,  and 
likewise  the  bevels,  of  the  hip  rafter.  G  indi- 
cates the  run  of  hip,  and  H  the  length  of  hip. 

The  following  is  a  useful  application  of  the 
steel  square:  On  the  left  of  diagram,  8  and  12 
inches  on  the  square,  cut  the  common  rafters;  if 


70 


PRACTICAL  USES  OF 


I  i 


it  rises  8  inches  in  i  foot  it  will  rise  lo  feet  8 
inches  on  i6  feet.  Using  J<-inch  to  i  foot  scale, 
place  the  square  on  the  line  (Fig.  35)  at  3  and  2, 


representing  the  12  feet  8  feet.  Now  slide  the 
square  up,  to  bring  4  on  the  line,  as  shown  by 
dotted  lines,  which  gives  10  feet  8  inches  rise  in 
16  feet,  which  may  be  stated  thus,— 12  : 8  ::  16  ft. 
:  ID  ft.  8  in.  It  will  be  seen  that  the  steel  square, 
as  a  mechanical  device,  will  solve  problems  both 
in  square  root  and  simple  proportion. 

Perhaps  the  following  examples  on  the  subject 
which  were  submitted  by  correspondents  to  "Car- 
pentry and  Building"  may  prove  useful  to  my 
readers,  as  they  contain  several  good  ideas  which 
are  worth  considering.     I  have  made  some  slight 


THE  STEEL  SgUARE 


»i 


changes  in  the  text  in  order  to  make  it  suitable 
to  these  pages,  but  this  does  not  affect  the  sub- 
ject-matter in  the  slightest:  In  the  lay-out  shown 
at  Fig.  36,  we  have  a  3x6  valley  rafter  and  find 


Fig.  36 

the  center  of  the  face  side  at  the  point  A,  where 
it  intersects  the  two  ridges,  as  shown  in  Fig.  ^-j, 
the  center  line  being  AB.  Now  measure  one- 
half  the  thickness  of  the  rafter— that  is,  ij^ 
inches  from  A,  which  gives  the  point  C.  Squar- 
ing across  gives  the  points  D  and  E.  Connect- 
ing the  points  A,  D,  and  E  gives  the  angle  cut 


i'i 


m 


P';1 


-  'i 


7i 


•    V. 


practilal  rsFs  op 


where  the  ridges  meet  at  I),  as  shown  in  sketch 
It  must  be  evident  that  if  C  is  raised  to  stand 
directly  over  B  we  get  the  angle  f  -  slope  of  the 
valley  rafter,  also  ti.c  angles  of  bevels.     In  the 


Fjg.  37 


diagram  shown  at   Fig.  37.  the  square  is  set  to 
show  the  method  of  getting  the  proper  cuts  and 
lengths.  AB  shows  the  horizontal  line  or  seat  of 
valley,  while  AC  shows  the  run  or  length  of  val- 
ley rafter.     CB  on  diagram,  Fig.  36,  shows  the 
rise  of  the  rafter.     If  the  distance  CB  be  used 
on  the  tongue  of  the  square,  and  AB.  the  seat, 
set  off  on  the  blade,  these  will  give  the  plumb 
and  bevel  cuts  of  the  rafter;   while  the  bevels 
shown  in  Fig.  ^j  give  the  cuts  for  sides  of  hip  or 
valley. 


k: 


VALLEY  RAFTER 


Fig.  3S 
A  plan  of  the  valley  rafter  as  laid  off  for  cut- 
ting bevels,  is  shown  at  Fig.  58. 


THE    >TEEL  sQUARE 


73 


Fi^'  39  shows  position  of  rafter  where  ridges 

meet. 

Fig.  40  shows  an 
elevation  of  the  val- 
ley rafter  in  its  proper 
position  on  the  wall 
plates.  VVc  will  sap- 
pose  that  the  rise  is 
9  inches  t'  the  foot 
run,  as  sli  v.^  w  n  .  in 
which   case    the     ly-  pio.  39 


-%*^^ 


J 


■  m 


Fia.  40 


74 


'it 


PRACTICAL  USES  OF 


pothenuse  or  line  of  rafter  is  ,5  inches  to  -foot 
run.    F.g.  4,  shows  tl,e  method  of  obtaining  tTe 


PlO.   41 

bevel.  Take  15  on  the  blade  and  r^  on  the 
tongue  of  the  square,  place  it  on  the  rafter  as 
shown  and  the  tongue  will  give  the  desired  bevel 
marking  along  the  blade.  In  order  to  frame  a 
rafter  against  two  ridge  boards  running  at  right 
angles,  draw  a  line  in  the  center  of  the  rafter 
and  reverse  the  square.  This  rule  works  on  all 
pitches. 

Again,  suppose  the  half-width  of  a  roof  having 
a  pitch  of  45°  is  10  feet,  and  that  an  adjoining 
roof  IS  one-third  pitch,  then  it  will  take  15  feet 
of  It  to  make  an  equal  rise.     By  the  conditions 
of  the  problem  we  then  have  a  rectangle  loxi. 
feet  by  which  to  get  the  length  of  the  valley 
rafter  sought.     A  line  drawn  diagonally  through 
this  rectangle   will  give   the   run  of  the  valley 


THE  STEEL  SQUARE 


75 


rafter.     Lay  off  at  right  angles  each  way  from 
the  diagonal  a  distance  equal  to  the  rise,  Fig.  42, 


Fig.  42 

and  connect  it  as  shown  in  the  diagram.  This 
will  give  the  length  of  the  valley  rafter.  Let 
fall  on  each  side  of  the  diagonal  a  perpendicular 
equal  to  the  half-width  of  the  rafter  terminating, 
at  the  sides  of  the  figure.  From  B  and  C  thus 
established  let  fall  perpendiculars  BA  and  CD 


»} 


i    t| 
1  "M 

A 

If 

"  M 


7« 


PRACTICAL  USES  OF 


from  the  line  of  the  valley  rafter.     Then  AB  will 

be  the  backing  or 
distance  above  the 
edge  of  valley  to 
set  the  jacks  for 
the  45°  pitch,  and 


CD  will  be   the 
height  above  the 
edge  of  valley  to 
set  the  jacks  for 
the  third    pitch. 
The  line  on  the 
plate  will  be  ob- 
tained as  shown 
in  Fig.  43,  using 
the  square  with 
lOon  the  tongue 
and    15   on    the 
blade. 

From  W.  H. 
Croker,  of  Oril- 
lia,  Ont.,  who  is 


}■  44 


i    J:  i 


THE  STEEL  SQUARE  77 

an  excellent  authority,  I  get  the  following  on  the 
same  subject:   Suppose  the  plan  of  che  plates  is 
AB,  Fig.  44,  in  any  given  building,  and  the  cor- 
responding rafters  A'B'.     Where  the  top  lines  of 
rafters  intersect,  marked  i  on  the  elevation,  drop  ' 
a  plumb  line  14  to  intersect  4-6,  made  at  an  angle 
of  45°,  and  passing  through  the  internal  angle  of 
the  plates.      At  any  point   eaveward  draw  2-3 
horizontal,  and  from  the  point  of  intersection  3 
drop  the  plumb  line  3-6,  and  from  where  it  inter- 
sects the  line  4-6  draw  6-5  parallel  to  2-3.     Make 
6-5  equal  to  2-3.     Then  a  line  drawn  as  shown  by 
5-4  will  be  the  plan  of  the  center  of  the  valley 
rafter.     One-half  of  the  thickness  of  the  rafter 
laid  off  on  each  side  of  4.5  will  determine  the 
relative  position  of  the  valley  rafter  to  the  plates 
In  order  that  the  student  of  this  work  may  be 
armed   with    the  proper  theory  underlying  the 
formation  of  hip  roofs,  I  submit  the  following 
which    IS   taken    from    Peter   Xlcholson,   whose 
methods  for  finding  "working  lines"  in  timber 
framing  have  never  been  excelled. 

Let  abed,  Fig.  45,  U,  the  plan  of  a  roof,  wy  the 
width  or  beam,  ix  the  height  of  the  roof,  zox 
and  i.'y  the  length  of  the  common  rafters-  to 
find  the  length  of  x\^v.  hip  rafters  from  the  data 
here  given  proceed  as  follows: 


W 


^^■-  .■  ii 


mmmm. 


I 


78 


f     r 


t 


PRACTICAL  USES  OP 


Bisect  each  end  of  the  plan  a6,  cd,  in  the  points 
y  and  4.  and  draw  the  plan  ^r  of  the  ridge  line. 


li.. 


THE  STEEL  SQUARE  ^^ 

Bisect  the  angles  at  a,  b,  c,  dot  the  roof,  by  the 
lines  as,  6s,   cm,  dm,    meeting  the  plan  of  the 
ndge  line  in   .  and  m  and   the  lines  as,  6s,  cm 
dm  are  the  plans  of  the  hip  rafters.     From  the 
pomt  where  the  plans  of  the  hip  rafters  meet  the 
ndge  Ime,  draw  a  perpendicular  to  each  of  the 
h.p  rafters,    and  set  the  height  ix  of  the  roof 
upon  each  perpendicular;  and  the  hypothenuse 
of  each  nght-angled  triangle  will  be  the  length 
of  each  hip  rafter.      Thus   find  th^  hip  rafter 
over  dm:    Draw  mz  perpendicular  to  ./...;  make 
^uz  equal  to  ix,  and  join  mz;  then  mz  is  the  length 
of  the  pnncipal  rafter  over  dm. 

The  hip  rafter  may  be  found  very  conveniently 
'n   the    following  manner:      Produce   iw  to    /• 
make  tt  equal  to  md,  and  draw  tx,  which  will  be 
equal  to  dz;   and  thus  the  remaining  three  will 
be  found      To  find  the  backing  of  the  hips,  draw 
^S  at  right  angles  to  md;  from  the  point  /as  a 
center  find  the  radius  of  a  circle  which  will  touch 
the  hne^.;    make  .^  equal  to  that  radius,  and 
jom^/and  ,c;   then  the  angle  .^y  is  the  angle 
of  the  back  of  the  rafter.     This  method  of  find- 
;ng  the  backing  of  hip  rafters  is  said  to  be  the 
-nvention  of  a  Mr.  Pope.     In  this  illustration  and 
'description  almost  every  possible  shape  of  hip 
roof  plan    is   involved,    and   from   it  hips  and 


i  ."fi 


•.a  . 

1 

J 

>i. 

w :  rm  \t: 


80 


% 


I  I 


PRACTICAL  USES  OP 


jacks  may  be  determined  with  their  lengths, 
bevels,  and  inclinations,  without  much  trouble! 
It  will  be  noticed  that  the  steel  square  is  not 
employed  in  this  description,  or  in  the  illustr^x- 
tion.  This  is  due  to  the  fact  that  Mr.  Nichol- 
son's works  were  published  long  before  the 
American  steel  square  came  into  general  use. 
T:  '1  illustration  is  the  most  complete  of  its  kind 
known,  and  this  is  partly  my  excuse  for  its 
reproduction  in  this  work. 


II     1 


ti 


UNEVEN    PITCHES 

Irregular,  uneven,  or  unequal  pitches,  are 
simply  different  pitches  in  the  same  roof. 
When  they  are  the  same  on  all  sides  and  the 
building  is  square,  the  hip.  or  valleys  run  in 
from  the  corners  at  an  angle  of  45  degrees, 
regardless  of  the  rise 
of  the  roof;  but 
should   one   side    be 

steeper  than  the  ad-     H 

jommg  side,   or   the  «H 

gables  be  of  different    "^ 

pitch  from  the  main 

roof,  then  the  hips  or 

valleys  depart    from 

the  45"  angle. 


FlQ.  46 


THE  STEEL  SQUARE  gi 

Fig.  46  shows  a  roof  plan  with  the  one-third 
pitch  on  the  main  part,  with  a  half-pitch  gable. 
The  seat  and  down  cuts  of  the  jack  and  com- 
mon rafters  remains  the  same  as  in  the  even- 
pitch  roof,  except  the  top  cut  of  the  jack. 

I  will  not  take  up  space  to  explain  this  cut 
at  length,  but  will  give  that  obtained  by  the 
square  as  follows:  Take  to  scale  the  length  of 
the  left  common  rafter,  on  the  blade  and  the 
run  of  the  right  common  rafter  on  the  tongue. 


CuToruerrJApr 


Fio.  47 


Blade  gives  the  cut  of  the  left  jack-z./<r^  versa 
for  the  right  Jack.  Figs.  47  and  48  illustrate 
these  cuts. 


in 

I 


CuroFRiSMr  ^t 
Jack 


Fio.  48 


Here  is  another  problem  that  comes  in  con- 


% 


i 


82 


i 

I 

h 
I 


i   ! 


t     ' 


PRACTICAL  USES  OP 


nection  with  the  uneven-pitched  roof.  Where  a 
projecting  cornice  is  desired,  with  planceer,  tae 
valley  will  not  rest  at  the  angle  of  the  plate,  but 
at  a  point  in  line  with  the  intersection  of  the 
cornice,  as  shown. 

This  necessitates  the  plate  on  the  steeper 
pitch  being  raised  as  much  as  the  difference  in 
the  rise  of  the  pitches  in  the  width  of  the  cor- 
nice. 

Thus,  if  the  cornice  be  18  inches  wide,  the  rise 
of  the  half-pitch  is  18  inches,  and  that  of  the 
one-third  pitch  is  12  inches,  a  difference  of  6 
inches.  Therefore,  the  proper  height  of  the 
plate  above  that  of  the  lower  pitch  is  6  inches. 

In  connection  with  the  above,  Mr.  Woods  gives 
the   following   explanations   and    diagrams    for 
finding  the  lengths  of  rafters  where  the  rises  in 
the  roof  are  of  different  heights,      hor  example 
we  will  suppose  the  main  gable.  Fig.  49,  to  be  24 
feet  wide  with  a  14-foot  rise,  and  th.  side  gable 
to  be  16  feet  wide  with  a  lo-foot-S-inch  rise      In 
a  case  of  this  kir^  it  is  better  to  let  one  of  the 
valleys  extend  on  up  to  the  ridge  board  of  the 
main  gable  and  let  the  other  valley  rest  against 
It  (the  long  valley).     But  how  to  locate  them  on 
the  square  is  the  main  question,     ist.  Place  the 
squares  as  shown.    On  square  No.  i  lay  off  the 


THE  STEEL  SQUARE 


83 


run  and  rise  of  the  wide  gable,  and  the  same  for 
the  narrow  gable  on  square  No.  2, 


2d.  By  connecting  the  run  and  rise,  as  shown 
by  the  diagonal  line,  on  each  of  the  squares 
will  be  the  length  of  the  common  rafter. 

3d.  Square  out  from  the  tongues  as  shown, 
till  they  intersect  at  A,  which  will  be  the  runs  of 
the  gables  or  of  the  common  rafters. 

4th.  Set  compass  at  B,  and  open  to  equal  the 


I 


m 


84 


PRACTICAL  USES  OP 


4   i 


I     I    I 


n 


u 


III 


i  I 


nse  of  the  narrow  .^ble  and  swing  to  the  blade 
of  -No.  ..and  square  in  to  the  .mmon  rafter, 
thence  run  an  imaginary  hne  parallel  to  the 
blade,  ami  where  it  intersects  the  tongue  estab- 
I'shes  the  pent  where  the  ridge  of  the  narrow 
Rable  d.cs  or  i„f.  r.ects  on  main  roof  and  which 
point  I  will  call  C. 

5th    A  line  drawn  from  A  to  C  represents  the 
run  of  the  short  valley,  and  by  extending  the  line 

on  to  the  blade  of  Xo..  establishes 'point  D 
from  wh.ch  to  H  represents  the  run  of  the  long 
valley,  and  these  lengths  taken  on  the  tongues 
as  shown,  and  connected  with  then  respective 
rises,  wdl  be  their  lengths. 
6th.  The    lengths  of  the  jacks  are  found  as 

shown   from   E    to    F.    which    I  trust    is     clear 

enough  without  further  explanation. 

The  cuts  and  bevels  are  all  contained  in  this 
diagram. 

GENERAL   ITEMS 

On  the  subject  of  roof  framing.  Mr.  Stoddard 
says:  I  have  given  some  thought  and  study  to 
roof  frammg,  and  have  concluded  the  square  is 
master  of  the  situation,  as  it  is  much  quicker  and 
less  hable  to  mistake  than  any  method  I  know     " 


i    1 


THE  STEEL  SQUARE  85 

Let  us  take  the  number  of  inches  the  roof  is  to 
rise  to  the  foot  on  the  tongue  and  one  foot  on 
the  blade  (which  is  the  rise  and  run  of  one  foot). 
If  the  building  is  14  feet  wide  at  a  7-foot  run, 

Fio.  50 
apply  sev,'n  times,  as  illustrated  in  Fig.  50.     To 
cut  octaKon    rafter,   apply    same    as    common, 
except  use  13  inches  in  place  of  12  inches  on 
blade,  hip  or  valley,  use  17  inches. 

To  cut  jacks,  if  you  wish  them  16  inches  apart, 
slide  the  square  up  to  16  inches;  if  20  inches, 
slide  up  to  20  inches,  and  so  on. 

The  r/si-  and  run  cut  on  rise  jrives  top  cut  and 
all  plumb  cuts;  the  run  gives  cut  on  plate  and 
all  level  cuts. 

The  side  cut  of  jacks  to  fit  hip,  and  valley  to 
fit  ridge,  etc.,  is  /c»g^//i  of  nifhr  and  run,  cut  on 
length. 

These  general  rules  apply  to  all  roofs,  and  this 
is  roof  framing  in  a  "nut-shell,"  although  it  may 
not  be  new,  original,  or  even  the  best. 

But  a  better  way  yet  is  to  take  rise  and  run, 
measure  across  and  get  length  of  rafter;  this 
gives  length  of  all  rafters  for  even  or  uneven 
pitches,  and  all  main  cuts,  which  is  a  very  impor- 


II 


3 


H -I 


MICROCOPY   HESOIUTION   TEST   CHART 

(ANSI  and  ISO  TEST  CHART  No    2) 


1.0 


1-25  iu 


2.5 


m 

1^       n..i^ 


Z2 
2.0 


1.8 


1.6 


^     ^IPPLIED  IIVMGE 


1653   East    Mam    Street 

Rochester.    New    York         1-1609        U'^A 

(716)    482  -  0300  -  Phone 

(716)   288  -  5989  -  Fa» 


tkt*2Mm 


i  1' 


.[,.! 


h 


'  f  I 


86 


PRACTICAL  USES  O^ 


tant  matter  and  saves  much  trouble  and  worry 
where  unequal  pitches  are  to  be  worked  out. 


Fig.  51 


To  illustrate  this  I  will  take  a  little  24-foot  cot- 
tag?,  one-third  pitch  hip  roof,  4-foot  deck  and 
gable  in  front.    See  cut,  Fig.  51. 


Fio.  53 

As  it  is  only  the  principle  involved,  for  con- 
venience in  illustrating  we  will  use  even  feet  as 
much  as  possible,  and  not  give  accurate  measure 


I 


■■■.?  ■•«Rfc'^-iasa 


.ss-7-«v 


THE  STEEL  SQUARE 


87 


ments  as  to  inches,  although  in  real  framing 
accurate  measurements  should  always  be  made. 

One-third  pitch  roof  rises  8  inches  to  the  foot. 
As  this  24  foot  house  has  a  4  foot  deck,  the  run 
of  common  rafter  would  be  10  feet,  as  the  rise  is 
6  feet  8  inches  and  the  run  10  feet,  the  length  of 
common  rafter  is  12  feet,  Fig.  52. 

As  the  run  of  the  hip  is  the  diagonal  of  10  feet 


Fig.  53 

or  14  feet,  Fig.  53,  and  the  rise  is  6  feet  8  inches, 
run  14  feet,  length  of  hip  is  15  feet  6  inches.  Fig. 

54.  ^^ 


Fig.  54 

If  the  jacks  are  to  be  16  inches  apart,  measure 
across  your  square  at  16  inches,  at  one-third 


't.T 


i    \ 


ii 


38 


PRACTICAL  USES  OF 


pitch,  and  you  have  19  inches,  Fig.  55,  length  of 
short  jack;  twice  that  length  is  length  of  second 


i'<,^\\iff\ 


M  I  I  I  I  I  1 

Fig.  55 

ones,  and  so  on;  or  divide  the  common  rafter 
into  the  number  of  jacks  required  and  get  your 
lengths  from  common  rafter. 

As  the  length   of  cur  of 
common  rafter  is  12  J/^CK 
feet  and  run  10  feet,  ,2 
place  the  square  on 
12  and  10;  cut  on  12  pio.  56 

for  bevel  of  jack  rafter,  Fig.  56. 


li  1  I  I  i^f-jii 


JLLLJJ 


Pig.  57 


Now.  as  the  front  gable  is  to  show  the  roof, 


THE  STEEL  SQUARE 


89 


divide  into  about  three  equal  parts,  allowing  for 
projections;  set  the  foot  of  valley  4  feet  6  inches 
from  center  of  build -ng,  as  it  runs  back  10  feet  to 
deck. 


Fig.  58 

The  run  of  valley  is  1 1  feet,  Fig.  57;  as  the  rise 
>s  o  feet  8  inches,  run  1 1  feet,  length  of  valley 
rafter  13  feet,  Fig.  58. 


L 


'''''■■■'  '^J^'  '  ' 


f-fe- 


Fig.  59 

As  the  rise  of  front  gable  is  6  feet  8  inclies  and 
run  4  feet  6  inches,  length  of  gable  rafter  8  feet, 

As  the  length  of  common  rafter  on  main  roof 


'\ 


i>. 


:',j!AOTsai-cifr-«f^'" 


S?^  !^?^tJ^ 


i  I 


MI   I  ' 


L  . 


90 


PRACTICAL  USES  OF 


is  12  feet  and  run  of  gable  4  feet  6  inches,  place 
the  square  on  length  and  run  cut  on  length,  and 


Fig.  60 


it    gives    sic-   cut  of  main   jack   to   fit  valley. 
Fig.  60. 


Fig.  61 


As  the  gable  rafter  is  8  feet  and  run  of  main 
roof  10  feet,  length  and  run  cut  on  length  gives 
side  cut  of  gable  jack.  Fig.  61. 

Most   workmen   who   have    followed    all    the 
examples  given  in  this  work  are  aware  that  the 
rise  of  the  valley  or  hip  taken  on  the  square  will 
give  the  seat  and  plumb  cuts,  but  to  cut  the  seat 
so  that  the  top  edge,  backed  or  unbacked,  will 
coincide  with  the  plane  of  the  common  rafter  is 
a  problem  that  many  are  not  so  sure  of;  but  go 
ahead  and  make  the  cut,  trusting  to  luck,  and  if 
it  doesn't  come  right,  block  up  or  cut  down  as 
the  case  may  be,  and  the  matter  is  dismissed  for 
the  time,  only  to  reoccur  on  the  next  job. 


THE  STEEL  SQUARE 


In  order  to  enable  the  workmen  to  get  positive 
results,  the  following  illustrations  and  text  are 
submitted;  they  have  appeared  befor?  in  a 
different  shape,  but,  as  I  stated  in  the  outset,  it 
is  my  intention  to  publish  in  thf"  work  every- 
thing that  in  my  judgment,  will  oe  of  service 
to  the  reader  and  that  is  in  anyway  connected 
with  the  use  of  the  steel  square. 

The  illustration  shown  at  Fig.  62  exhibits  the 
position  of  a  hip  or  valley  rafter  when  the  roof 
is  of  equal  pitch.  A,  being  at  the  corner  of  plate 
for  either  a  hip  or  valley.  If  the  former,  it 
sides  will  intersect  the  edge  of  the  plate  at  B  and 
B,  or  at  C  and  C,  for  the  latter. 

The  distance  from  A  to  B  and  B,  or  C  and  C, 
is  always  equal  to  the  diagonal  of  a  square  with 
sides  equal  to  one-half  the  thickness  of  the 
rafter.  If  the  rafter  be  2  inches,  then  the 
distance  will  be  lA  inches.  BC  (along  the  side 
of  the  rafter)  's  equal  to  the  thic'  of  the 

rafter,  and  this  measurement  taken  square  out 
from  the  plate  at  BC,  and  by  transferring  the 
center  as  a,  will  give  the  different  positions  of 
the  seat  cuts  with  that  of  the  common  rafter. 

Now  passing  up  to  the  common  rafter,  DE  is 
the  depth  desired  from  the  plate  to  the  top  edge 
of  the  rafter. 


1 4 


?t 


PRACTICAL  USES  OF 


li 


M 


i  1 1 


ii 


Fig.  62 


fftm<i*.. 


THE  STEEL  SgUARE 


93 


Passing  on  to  the  hip.  If  the  same  is  not  to  be 
backed,  DE  will  be  the  same  as  of  the  common 
rafter,  but  the  seat  cut  will  extend  to  <'/on  the  a 
line,  but  if  to  be  backed,  dc  will  be  the  depth 
and  will  equal  DE  when  backed  where  the  sides 
of  the  rafter  pass  over  the  edge  of  the  plate  at 
BB.  The  face  view  of  the  seat  would  show  as 
per  the  shaded  part  of  section  at  V. 

Passing  up  to  the  valley,  the  depth  above  the 
plate  at  A  will  be  de,  backed  or  unbacked. 
The  seat  cut  will  extend  to  U  at  the  sides,  and 
will  intersect  the  plates  at  CC. 

If  to  be  backed,  DE  will  be  the  depth  above 
plate  at  CC.  The  shaded  part  of  section  at  G, 
shows  the  face  of  the  seat  cut  in  case  the  un- 
der side  of  the  tail  is  backed;  however  this  is 
illy  omitted.  The  reader  will  understand 
like  letters  represent  like  measurements, 
oei"  J  over  the  plate  at  B  or  C,  diwd  de 
over  A.  The  solid  lines  represent  the  rafters 
when  not  backed,  and  the  broken  or  dotted  lines 
when  backed.  From  this  it  will  be  seen  that  the 
backing  of  a  hip  or  valley  is  obtained  by  setting 
off  one-half  of  its  thickness  on  the  seat  bevel,  aS 
at  c'E,ord  D. 

In  Fig.  63  is  shown  the  plan  of  a  hip  and  com- 
mon rafter  in  place,  also  an  elevation  of  same 


■^i&j''y^£!^m¥^'^ms!m^i^^^js^ 


OiNi^i^ '*".i>Wyp^ 


f  i 
%  i 


fe  f  i 


t  I 


94 


PRACTICAL  USES  OF 


with    the   hip    swung   parallel   to   the  common 
rafter,  AC  and  Ali  being  their  respective  lengths. 


Pig.  63 


A  method  of  laying  out  a  hip  roof  and  making 
a  cardboard  model  for  same  was  published  in 
"Carpenter,"  some  time  ago  by  Mr.  Henry  D. 
Cook,  of  Philadelphia,  Pa.,  and  which  is  repro- 


mm^  T^i  ^^mwmm<^' 


;:$k?*iKi?»vi;^     h'>t'^*^Ml^ 


THE  STEEL  SQUARE 


95 


cluced  here,  as  I  think  it  worthy  of  a  place  in  the 
present  work. 

The  most  simpk  fornj  of  hip  roofs  is  tnat 
where  the  ground  phm  of  the  building  makes 
right  angles.  In  the  ordinary  hip  roof  but  little 
constructive  skill  is  required,  the  onlj  points 
requiring  particular  attention  are  in  finding  the 
proper  lengths  and  side  cuts  of  jack  rafters,  and 
those  can  be  made  quite  simple.  To  do  this,  sup- 
pose we  get  a  piece  of  cardboard  and  commence 
laying  down  the  ground  plan  of  a  building,  which 
we  will  represent  by  letters  A,  B,  C,  1 ).  Next  lay 
out  the  elevation  of  one  pair  of  rafters  B,  E,  C. 
shown  at  A,  Fig.  64.  Next  lay  down  the  seat  of 
the  hip  at  an  angle  of  45  ;  on  each  side  set  off 
half  thickness  of  hip  which  draw  parallel  with 
cent,  line  AF;  from  the  seat  line  AF  square 
out  from  F  to  G;  make  GF  in  A,  Fig.  64,  equal 
HE  in  the  same  diagram,  and  square  out  lines 
IJ  and  KL,  and  join  AG,  which  gives  the 
back  line  of  the  hip  rafter;  next  layoff  the  seqts 
of  the  jack  rafters  on  line  AD,  and  make  MN 
equal  the  given  rafter  BE  or  CE,  and  join 
I),  N,  on  each  s^de  of  which  set  off  half  thick- 
ness of  the  hip;  next  square  over  the  seat  lines 
of  the  jacks  on  line  iD,  and  let  them  cut  the 
seat  of  hip  as  represented  on  the  plan;  then  with 


■  m 


|.!l'' 


'i  IT  a 


W^^--'!im^W^WM'JP: 


<j6 


PRACTICAL  USICS  OF 


i  i 


your  dividers  take  I)  as  center  and  M  as  radius, 
and  strike  the  curve  line  cutting  at  O,  make  ON 


Fig.  64 

equal  MN,  and  m-ke  OP  equal  OD,  and  PN 
equal  HE,  in  \,  1  g.  64,  and  lay  off  your  jacks  on 
line  OD.  Now  with  one  point  of  your  dividers 
press  it  through  at  center  point,  A,  also  at  F; 
from  these  impressions  on  hack  side  of  card,  and 
with  a  sharp-pointed   knife,  cut  partly  through 


t   : 

»   i 


i 


T.  .*MiBW^^vmmsm*iKatx^mmmj^^'rwts:^i3ltkarik 


THE  STEEL  S<jUARE 


97 


the  cardboard  only.  And  from  A  to  G,  and  (i  to 
V,  on  face  side:  cut  clear  through  the  card;  the 
hi  I )  is  now  ready  to  raise;  the  line  AF  will  form 
a  hinge;  but  before  doing  this  turn  '  our  atten- 
tion to  center  line  at  M;  on  this  lin»  /ou  will  cut 
through  from  M  to  N,  and  on  line  om  I)  to  (  ■ 
you  will  cut  clear  through,  from  O  to  I*  will  also 
have  to  be  cut  clear  through,  and  on  line  from 
V  to  X  must  be  c.i    through. 

Next,  on  line  NO,  on  face  side,  cut  partly 
through  only,  as  this  must  form  a  hinge  when  you 
begin  to  turn  up  the  work;  next  on  center  line 
ND,  on  face,  cut  partly  through  only,  and  on  line 
from  M  to  D  cut  partly  through  from  the  brick. 
You  are  now  ready  to  raise  this  section  of  the 
roof.  Now,  at  point  X,  commence  to  ruse  up 
your  work,  and  you  will  notice  as  you  i  e  the 
point  N  the  point  P  will  fall  and  will  s.^nd  at 
letter  F,  and  the  point  at  O  will  .vvuig  around 
and  stand  on  line  CD.  ou  will  I'.en  have  a 
perfect  model  of  one  corner  of  the  roof  of  the 
building,  and  at  the  same  time  the  hip  will  show 
its  perfect  backing. 

Next  come  to  AG  and  raise  or  turn  up  the 
hip,- and  see  how  nicely  it  agrees  with  the  other 
portion  of  the  roof;  two  hips  will  come  together 
when  points  X  and  G  will  meet. 


:|  .'' 


o8 


PRACTICAL  USES  OF 


You  can  go  still  fartner  by  pressing  the  point 
of  your  dividers  through  points  B  and  C,  and 
from    these    impressions    on    back    cut    partly 
through  the  card  only,  and  on  lines  BE  and  EC 
you  will  cut  clear  through;  you  are  now  ready  to 
raise  the  given  rafters,  which  should  stand  at  a 
right  angle  with  the  cardboard,  when  it  will  be 
seen  that  the  backs  of  the  given  rafters,  also  the 
jacks  ar:l  the  hip.  all  agree  and  will  all  be  on  a 
straight  line. 

Backing  the  hip  at  any  point  on  the  seat,  say 
R,  square  out  a  line  cutting  line  AB  at  S;  take 
R  as  center,  and  a  circle  cutting  back  of  hip,  also 
center  of  seat  at  T.  join  ST,  which  gives  bevel 
2  for  the  backing.     The  bevels  for  the  plumb  and 
foot  cuts  of  the  hip  are  seen  at  3  and  4,  and  the 
bevels  for  the  plumb  and  foot  cuts  of  the  given 
and  all  the  jack  rafters  will  be  seen  at  5  and  6 
and  the  bevel  for  the  side  cut  or  face  of  the 
jacks  will  be  seen  at  7. 

To  cut  the  hip  so  as  to  fit  against  the  ridge 
you  will  notice  the  points  J  and  L,  in  B,  crossing 
the  line  of  the  hip  at  the  point  L,  will  be  the 
longest  point  of  the  hip,  and  at  point  J,  square 
over  the  back  of  hip  and  marked  on  the  other 
side,  gives  the  shortest  cut.  As  all  measure- 
ments  are  taken  from  the  center,  one-half  thick- 


THE  STEEL  SQUARE 


ll^ 


99 


ness  of  the  ridge  will  yet  hav€  to  be  taken  off  at 
the  upper  end. 

The  construction  of  a  model  roof  in  card- 
board, after  the  method  as  advanced  in  the  fore- 
going, will  materially  assist  the  workman  in 
grasping  the  true  principles  of  roof  framing,  and 
enable  him  to  understand  the  reasons  "why  and 
wherefore"  all  the  necessary  lines  for  laying  out 
such  a  roof  as  described  may  be  obtained  by  an 
intelligent  application  of  the  steel  square. 

Before  leaving  the  subject  of  ordinary  roof 
framing,  I  wish  to  reproduce  from  "Modern  Car- 
pentry" an  extract  from  an  English  source, 
which  deals  with  roof  framing  somewhat  differ- 
ent to  American  practice.  I  have  changed  and 
amended  the  text  somewhat  in  order  to  make  it 
more  easily  understood  by  American  workmen. 
In  order  to  give  a  general  idea  of  the  use  of 
the  square  there  is  herewith  appended  a  few 
illustrations  of  its  applications  in  framing  a  roof 
of,  say,  one-third  pitch,  which  will  be  supposed  to 
consist  of  common  rafters,  hips,  valleys,  jack 
rafters,  and  ridges.  Let  it  be  assumed  that  the 
building  to  be  dealt  with  measures  30  feet  from 
outside  to  outside  of  wall  plates;  the  toe  of  the 
rafters  to  be  fair  with  the  outside  of  the  wall 
plates;    the  pitch    being  one-third   (that  is,  the 


ik 


lOO 


PRACTICAL  USES  OP 


I 


l| 


roof  rises  from  the  top  of  the  wall  plate  to  the 
top  of  the  ridge  one-third  of  the  width  of  the 
building,  or  lo  feet);  the  half-width  of  the  build- 
ing being  15  feet.  Thus,  the  figures  for  working 
on  the  square  are  obtained;  if  other  figures  are 
used,  they  must  bear  the  same  relative  propor- 
tion to  each  other. 

To  get  the  required  lengths  of  the  stuff, 
measure  across  the  corner  of  the  square,  from 
the  lo-inch  mark  on  the  tongue  to  the  15-inch 
mark  on  the  blade.  Fig.  65.    This  gives  18  feet 


I  ;r 


Fig.  65 

as  the  length  of  the  common  rafter.  To  get  the 
bottom  bevel  or  cut  to  fit  on  the  wall  plate,  lay 
the  square  flat  on  the  side  of  the  rafter.  Start, 
say,  at  the  right-hand  end,  with  the  blade  ^f  the 
square  to  the  right,  the  point  or  angle  of  the 
square  away  from  you,  and  the  rafter,  with  its 
back  (or  what  will  be  the  top  edge  of  it  when  it 
is  fixed)  toward  you.  Now  place  the  15-inch 
mark  of  the  blade  and  the   lo-inch  mark  of  the 


THE  STEEL  SQUARE 


xoi 


tonofue  on  the  corner  of  the  rafter— that  is, 
toward  you— still  keeping  the  square  laid  flat, 
and  mark  along  the  side  of  the  blade.  This 
gives  the  bottom  cut,  and  will  fit  the  wall  plate. 
Now  move  the  square  to  the  other  end  of  the 
rafter;  place  it  in  the  same  position  as  before  to 
the  i8-foot  mark  on  the  rafter  and  to  the  lo-inch 
mark  on  the  tongue  and  the  15-inch  mark  on  the 
blade;  then  mark  alongside  the  tongue.  This 
gives  the  top  cut  to  fit  against  the  ridge.  To 
get  the  length  of  the  hip  rafter,  take  15  inches 
on  the  blade  and  15  inches  on  the  tongue  of  the 
square,  and  measure  across  the  corner.  This 
gives  2ii'ii  inches.  Now  take  this  figure  on  the 
blade  and  10  inches  on  the  tongue,  then  meas- 
uring across  the  corner  gives  the  length  of  the 
hip  rafter. 


Fig.  66 

Another  method  is  to  take  the  17-inch  mark 
on  the  blade  and  the  8-inch  mark  on  the  tongue 
and  begin  as  with  the  common  rafter,  as  at  Fig. 


)  "in 

m 


-.il 


9 "  m 


loa 


PRACTICAL  USES  OP 


66.    Mark  along  the  side  of  the  blade  for  the 
bottom  cut.     Move  the  square  to  the  left  as 
many  times  as  there  are  feet  in  the  half  of  the 
width  of  the  building  (in  the  present  case,  as  we 
have  seen,  15  feet  is  half  the  width),  keeping  the 
above-mentioned  figures   17  and  8  in  line  with 
the  top  edge  of  the  hip  rafter;   step  it  along  just 
the  same  as  when  applying  a  pitch  board  on  a 
stair-strmg.  and  after  moving  it  along  ,5  steps, 
mark  alongside  the  tongue.    This  gives  the  top 
cut  or  bevel  and  the  length.    The  reason  17  and 
8  are  taken  on  the  square  is  that  12  and  8  repre- 
sent the  rise  and  run  of  the  common  rafter  to  r 
foot  on  plan  while  17  and  8  correspond  with  the 
plan  of  the  hips. 

To  get  the  length  of  the  jack  rafters,  proceed 
in  the  same  manner  as  for  common  or  hip  rafter- 
or  alternately  space  the  jacks  and  divide  the 
length  of  the  common  rafter  into  the  same  num- 
ber of  spaces.  This  gives  the  length  of  each 
jack  rafter. 

To  get  the  bevel  of  the  top  edge  of  the  jack 
rafter.  F,g.  67.  take  the  length.  ,43^^  inches  of  the 
common  rafter  on  the  blade,  and  the  run  of  the 
common  rafter  on  the  tongue,  apply  the  square  to 
the  jack  rafter  and  mark  along  the  side  of  the 
blade;    this  gives  the  bevel  or  cut.     The  down 


l^bifa 


n 


THE  STEEL  SQUARE  ,03 

bevel  and  the  bevel  at  the  bottom  end  ore  the 
same  as  for  the  common  rafter. 
To  get  the  bevel  for  the  side 
of  the  purlin  to  fit  against  the 
hip  rafter,   place  the   square 
flat  against    the  side   of    the 
purlin,  with  8   r'nches  on   the 
tongue  and  14.)^  inches  on  the 
blade,  Fig.  68.  Mark  alongside  C 
of  the  tongue.     This  gives  the 
side  cut  or  bevel.     The   14  >^  f.g.  67 

inches  is^the  length  of  the  common  rafter  to  the 
I -foot  run,  and  the  8  inches  rep- 
resent the  rise. 


i'  n  1,1 1 1 J 

For  the  edge  bevel  01  purlin,  lay  the  square 
flat  agamst  the  edge  of  purlin  with  12  inches  on 
the  tongue  and  14H  inches  on  the  blade,  as  at  Fig. 
'^9.  and  mark  along  the  side  of  the  tongue.  This 
f^'ves  the  bevel  or  cut  for  the  edge  of  the  purlin. 


i  .1 
I 


I04 


PRACTICAL  USES  OP 


The  rafter  patterns  must  be  cut  half  tl  j 
thickness  of  ridge  shorter,  and  half  th*:  thickness 
of  the  hip  rafter  allovi-ed  off  the  jack  rafters. 

A  few  remarks  regarding  the  backing  of  hi[) 
rafters  and  the  getting  of  the  pioper  lengths  of 
jack  rafters,  or  'cripples"  as  they  are  called  in 
some  sections  of  the  country,  and  1  have  done 
with  ordinary  roof  framing  for  the  present. 

I  have  shown  in  several  instances  how  the 
lengths  of  jack  rafters  and  their  bevels  may  be 
obtained,  but  I  have  not  specially  shown  how 
these  results  are  ol,taincd,  so  will  devote  some 
little  space  now  to  this  purpose.     Let  us  suppose 

AB  and  BC,  Fig.  70,  to  be 
hips,  and  AD  and  CD  val- 
leys laid  out  from  any 
particular  plan,  then  the 
jacks  cutting  in  between 
valley  and  hip  may  be  laid 
out  as  shown  at  FE;  the 
bevels  shov/ing  the  angles 
of  the  cuts,  the  plumb  cuts 
being  the  same  as  for 
common  rafters.  The 
bevel  at  E  shows  the  side  cut  against  the  hip, 
and  the  bevel  at  F  the  side  cut  against  the  valley 
rafter. 


Fic.  70 


THE  STEEL  SQUARE 


105 


Another  way   to    determine    the    length    of 
jack  rafters  is  given  as  follows:    On  the  ste  1 
square,  take   i?  inches  on  the  blade  and  the  rise 
of  the  roof,   12-foot   run,  on   the  tongue,  and 
measure   the   distance  across.     This  length  in 
inches,  multiplied  by  the  number  of  f<;et  the  jack 
ratters  are  to  be  on  centers,  will  give  the  required 
length  in  inches.     For  example,  if  the  roof  rises 
II  inches  per    foot  run,  measure  the   distance 
from  II  on  the  tongue  to  12  on  the  blade  of  the 
square,   which  is  16^  incl.es.     Now,  supposing 
the  jack  rafters  to  be  16  inches,  or  i}i  feet  on 
centers,  we  have  i6>'x  1^  =  21^   inches,  which 
is  the    difference   in   the  lengths   t  :   the    jack 

rafters. 

The  lengths  may  also  be  found  by  first  getting 
the  length  of  the  common  rafter  in  inches  for  a 
12-inch  run,  and  multiply  this  by  the  distance  in 
inches  the  jack  rafters  are  to  be  from  center  to 
center,  and  divide  the  result  by  12.  This  gives 
the  difference  in  the  length  of  jack  rafters  in 
inches.  For  example,  if  the  rise  is  12  inches  and 
the  run  is  12  inches,  the  run  of  the  rafter  is 
nearly  17  inches.  Now,  17  multiplied  by  28  and 
divided  by  12  gives  39-3  inches.  This  is  the 
differ  .ICC  in  the  length  of  the  jack  rafters  for  a 
one-half  pitch  roof  where  the  jacks  are  28  inches 


io6 


ii     I 


PRACTICAL  USES  OF 


from  centers.     This  rule  will  work  on  any  pitch 
of  roof. 

^^  Mr.  Hicks  pives  the  following'  rule,  in  his 
"Euilder's  Guide,"  for  obtaininjr  the  len^jths  of 
jacks,  which  is  somewhat  similar  to  that  already 
shown:  Take  the  run  of  common  rafter  on  the 
blade,  .2  inches,  and  the  len^nh  lJ^,\  inches,  on 
the  tongue,  and  lay  a  straight  edge  across,  as 


L 


"'""'' 


Fig.  71 


shown  in  Fig.  71.  Space  the  jacks  on  the  blade 
of  the  square,  which  represents  the  run  of  com- 
mon rafter,  and  measure  perpendicularly  from 
the  tongue  to  the  straight  edge  on  the  line  of 
each  jack  for  their  length. 

For  cutting  jacks  for  curved  roofs,  while  not 
exactly  within  the  scope  of  the  steel  square,  yet 
the  bevels  may  be  laid  off  by  that  instrument  as 
the  reader  will  no  doubt  discover;  so  I  give  here 
Mr.  Hicks'  method  of  determining  th  T lengths. 


THE  STEEL  SQUARE 


107 


The  curvature  of  these  rafters  will,  of  course,  be: 
governed  by  the  position  th<:y  ocf  upy  with  rela- 
tion to  the  hips.  The  method  offered  is  not 
new  by  any  means,  but  is  presented  in  a  manner 
easily  to  be  understood  by  the  ordinary  work- 
man. Let  us  suppose  AI), 
rig.  72,  to  be  the  run  of 
the  common  rafter,  DE 
the  rise.and  AE  the  length 
and  work  line.  To  find 
the  length  of  jack  set  off 
the  run  of  jack  AB  and 
square  up  the  rise  BC  to 
the  work  line  of  the  com- 
mon rafter;  then  AC  is 
the  length  of  jack  on  the 
work  line.  This  method* 
IS  very  simple,  yet,  as  it  is 
a  new  and  novel  way  of  finding  the  length  of  jack 
rafters,  it  will  be  well  to  point  out  a  common  mis- 
take which  the  inexperienced  might  chance  to 
make.  Bear  in  mind  that  A  E  is  the  length  of  com- 
mon rafter.  BC  is  not  the  length  of  jack,  as  some 
might  suppose,  but  the  rise  of  jack;  AC  is  the 
length  of  jack.  The  down  bevel  is  the  same  as 
that  of  the  common  rafter.  To  find  the  bevel 
across  the  back,  set  off  from    D  the  length  of 


Fig.  72 


'  sl 


ll 


i  if 

1 


if 


JjyuJ 


io8 


PRACTICAL  USES  OF 


common  rafter  to  F,  and  connect  F  with  A, 
which  shows  the  work  iine  of  the  hip.  Now 
continu.-  the  hne  BC  to  the  work  line  of  the  hip, 
and  the  bevel  at  G  will  be  the  bevel  across  the 
top  of  jack.  BG  is  also  the  length  of  jack,  and 
wi!l  be  found  to  be  the  same  as  AC. 

When  the  bevel  of  the  jacks  is  known  all  that 
is  necessary  is  to  square  up  the  rise  of  each 
jack  from  the  base  line  of  common  rafter 
AD  to  the  work  line  AE,  and  take  the  length 
from  A  to  the  point  where  the  rise  of  each 
jack  joins  the  work  line  of  common  rafter,  as 
shown. 

In  connection  with  hip  rafters  for  curved  roofs, 
it  may  be  well  at  this  point  to  depart  from  the 
course  pursued  so  far  in  the  making  of  this  book 
and  give  the  ordinary  lines  fcr  laying  out  such 
work  without  using  the  square  for  the  purpose. 
We  will  suppose  the  lines  A  and  B,  Fig.  73,  to 
represent  the  common  rafter  for  a  curvilinear 
roof,  let  B  represent  the  co.nmon  rafter,  and 
C  the  valley  rafter.  In  plan  and  profile,  respect- 
ively, the  curves  of  common  rafters  being  given, 
first  determine  the  seat,  or  base  line,  of  valley 
rafter,  which  in  roofs  of  this  kind  is  curved.  To 
illustrate,  the  common  rafters  were  cut  and  put 
up  on  both  sides  of  valley  strips,  tacked  on  one 


THE   STEEL   SgUARE 


109 


sulc  parallel  with  eav<;  or  ridge,  and  the  same 
number  of  strips  on  the:  other  side,  tacked  in  the 
same  manner  and  at  the  :  ame  verti'  al  hcij^ht;  it 


Fir,.    73 

is  evident  that  their  intersections  would  repre- 
sent the  line  of  valley.  Therefore,  the  curves  of 
common  rafters  being  drawn,  divide,  for  exam- 
ple, B  into  any  convenient  number  of  parts,  and 
through  the  points  thus  determined  draw  hori- 
zontal  lines.     The  lengths  of   these   lines   are 


i  ii 

!  11 

i  «  j 

S  .1 

5  .i.ij 


4,  -■? 


m 


*! 


no 


PKAlTICAL  USES  OP 


det(.'rmin<'(l  l)y  when-  they  cut  th 
ift 


('  curve  of  com- 


mon rafter,  and   art;  s«,t  off   on   correspond 
rafter  in  plan.     Then  draw  1 


ponding 
ines  parallel  with 


eaves  or  ridye,  as  shown  in  the  sketch,  and 
where  these  lines  intersect  is  the  base  line  C  of 
valley  rafter.  The  same  points  of  intersection 
project  on  corresponding  lines  of  profile,  giving 
the  line  of  valley  rafter. 

To  obtain  the  lateral  curve  of  valley  rafter, 
take  distance  on  outline  of  valley  rafter,  as  made' 
by  .he  horizontal  lines  already  described,  and 
draw  lines,  as  shown,  between  X  and  Z  at  the 
right  of  the  sketch.     Draw  a  vertical  line  cutting 
all  these  lines,  and  set  off  on  each  corresponding 
line  the  same    lateral  distance  as  between   the 
straight  and  curved  base  line  in  the  plan.     To 
find  the  top  bevel  of  valley  rafters,  let  the  thick- 
ness be  as  shown  at   D  in  plan.      Project   the 
length  of  bevel  by  half  the  thickness  on  curved 
line  in  profile.     Project  the  length  thus  found  on 
corresponding  line  on  upper  face,  as  shown  by 
XY  at  right  of  the  sketch.     From  the  two  points 
thus  determined  draw  a  line,  which  will  be  the 
true  bevel. 

In  connection  with  curved  roofs,  the  following 
is  offered  as  being  a  good  method  of  getting  the 
side  bevels  and   lengths   for  jacks  in   a  hipped 


'^'^'i^iSf^^i^zmrmf. 


-■■■i.  ■ 


hk'  a 


( 

r 

/ 

F 

e 

.y 

THE  STEEL  SQUARE  ,„ 

roof:  Let  CB  be  the  top  line  of  one  of  the  com- 
mon rafters.  In  the  diagram,  Fijr.  74,  CHU  \^ 
supposed  to  stand  upright  on  rise  HC  In  shape 
all  the  jacks  must 
he  some  parf;of  the 
length  of  th<;  com- 
mon rafter  meas- 
ured from  point  C. 
On  one  common 
rafter   lying   on    a  F";.  74 

flat  surface  with  marked  run  and  rise  must  be  laid 
off  all  the  jacks  showing  vertical  cut  also  long 
and  short  top  edges  opposit.-  to  each  other.     On 
run  make  CH  equal  to  F£G.  long  seat  line  of  this 
jack.    At  right  angles  to  run  draw  H  K.  the  verti- 
cal  cut.     Then  CK  is  the  long  top  etlge  of  this 
jack.    I- or  the  opposite  short  top  edge  :.  aw  a  line 
parallel  to  the  vertical  cut  at  a  distance  back  equal 
to  IG  taken  from  seat  of  jack.     On  top  face  of 
jack  mark  the  side  bevel  from  end  of  long  edge  to 
end  of  short  ^.\^^.     It  ,s  evident  that  when  jack 
CK  stands  upright  over  its  seat  its  bevelled  top 
nd  will   fit   against   the  hip  face  which  stands 
over  BC,  because  long  top  edge  of   uy-k  stands 
over  long  seat   line,  and  short   top  edne  of  jack 
>Un(is  over  short   seat  line;    only  if  ABC  is  an 
angleof45MoesFG  equal  thickness  of  jack.    For 


-I 


112 


PRACTICAL  USES  OP 


1  1 

I 


:   -  I        ; 

:■    I 


each  jack  the  side  bevel  will  be  different,  but  can 
be  obtained  in  this  manner. 

Before  leaving  the  subject  of  hip,  valley,  and 
jack  rafters  with  regard  to  their  lengths  and 
bevels,  I  think  it  will  be  in  the  interest  of  my 
readers  to   reproduce   a  system    of    lines   first 


K  '                  A 

w 

[ — r 

\\.      *2^y 

^ 

%e  ^^'^  M 

^ 

X 

\ 

\ 

\ 

^ 

X                  "'1        - 

M 

\          \ 

'\          /-\ 

i^ 

'  \ 

f 

J           /\ 

V 

\ 

\ 

rt 

^ 

\ 

\- 

/.     A/ 

-7^      . 

/.  /I 

M 

/ 

F 

f 

\ 

N 

Fir,.  75 


invented  by  Peter  Nicholson,  and  simplified  by 
Mr.  Smith  and  published  in  his  "Architect,"  a 


THE  STEEL  SQUARE 


"3 


book  that  at  one  time  had  a  deserved  popularity. 
While  the  book  is  now  seldom  spoken  of,  this 
system  of  lines  has  been  made  use  of  by  nearly 
all  the  late  writers   on  constructive  carpentry, 
with  greater  or  lesser  elaboration.     On  Fig.  75, 
the  plan   1,1,1,1,1  represents  the  outside  plate; 
2,2,  the  ridge  line;  3,3,3.3-3.3.  the  jack  rafters  of 
hip  and  valley;   4,4,  the  side  bevel  of  jacks  and 
the  length  of  jack  from  corner  of  plate  and  ridge 
to  side  of  hip  and  valley;  5,  bevel  at  head  of  hip 
and  valley;  6,  bevel  at  foot  of  hip  and  valley 
rafter;    7  is  a  common  rafter;    8,  the  bevel  at 
head  of  common  rafter,  is  the  down  bevel  for  all 
jacks  on  hips  and  valleys;   9,9  is  the  length  of 
hip  and  valley  rafter;    10  is  the  method  of  get- 
ting the  bevel  of  back  of  hip.     Draw  a  line  at 
right  angles  with  base  line  of  hip,  then  set  one 
foot  of  the  dividers  where  this  line  crosses  the 
base  line,  and  the  other  where  it  crosses  the  hip- 
rafter  line,  and  set  the  same  distance  on  the  base 
line,  and  draw  lines  from  that  point  to  the  plate 
each  way,  which  gives  the  bevel  for  hip,  and, 
turned  the  other  way  up,  it  gives  the  hollow  for 
the  back  of  the  valley.     Line  from  a  to  d  h  the 
length  of  hip  and  valley  dropped  down  to  get  the 
length  of  jacks.     Lengths  and  bevels  of  all  hips 
and  valleys  the  same  in  same  roof  of  same  pitch. 


•  -i 


114 


PRACTICAL  USES  OF 


^    ^i 


Fig.  76  is  a  plan  for  framing  a  valley  in  a  roof 
where  one  side  is  much  steeper  than  the  other, 


Fig.   76 

as,  for  instance,  one  side  rises,  say,  to  feet  in  8 
feet,  i-i  is  the  wall  line;  2-2  is  the  ridge  line;  3 
is  the  valley  rafter;  4  is  the  bevel  at  the  foot;  5 
is  the  bevel  at  the  head;  6  is  the  bevel  of  the 
jacks  on  the  lowest  pitch,  also  the  length  of 
same;  7  is  the  bevel  of  and  length  of  jacks  on 
the  steep  side;  g  is  common  rafter  on  the  lower 
pitch;  10  is  the  down  bevel  on  jacks  of  each  side; 
1 1  is  the  height  of  roof;  12  the  base  line  of  val- 
ley.    The  rafters  will  not  matcl.  on  the  valley  as 


THE  STEEL  SQUARE 


"5 


on  an  equal-pitch  roof,  as  in  Fig.  75.  It  will  be 

seen  that  it  will  take  seven  jacks  on  the  steep 

side,  while  it  requires  only  four  on  the  other 
side,  but  the  bevels  will  all  fit. 


BACKING    HIP    RAFTERS 

A  writer  on  Building  Construction  has  said: 
"In  America  there  are  more  hips  'backed'  in 
books  and  papers  than  there  are  in  houses." 
Unfortunately  this  is  too  true.  Most  workmen 
ne\er  think  of  backing  a  hip;  they  put  in  the 
timber  just  as  it  comes  from  the  lumber  yard, 
with  the  exception  of  cutting  the  bevels  for  point 
and  heel  of  rafters.  This  is  all  wrong.  All  hips 
should  he  backed,  in  order  to  get  a  good  strong 
and  nearly  perfect  roof.  When  the  hip  is  thin, 
being  no  more  than  two  inches  thi'^k,  it  is  not  so 
bad,  yet  it  ought  to  be  backed;  bui  when  the  hip 
is  three  or  more  inches  thick,  then  under  no 
circumstances  should  backing  be  omitted. 

In  the  present  volume,  as  well  as  in  the  former 
one,  I  have  shown  some  rules  for  getting  baciv- 
ing  for  hips,  but  in  order  to  have  the  principle 
well  understood  I  present  a  few  more  examples 
showing  how  the  angles  may  be  obtained. 

The  method  shown  at  Fig.  'j'j  is  a  quick  one, 
and   a  correct    one    if    the    measurements    are 


i    ■ '  rl 


-p  I 


i  WM 


ate 


Ji 


i 

I  ll 


ii6 


PRACTICAL  USES  OP 


exactly  taken.     The  diagram  explains  itself  and 
requires  no  description.     When  the  horizontal 


Fig.  77 

or  bottom  cut  for  a  hip  rafter  has  been  obtained, 
take  one-half  the  thickness  of  the  rafter  and 
measure  back  from  the  toe  or  point  toward  the 
heel.  This  will  give  the  point  on  the  side  of  the 
rafter  i-c  gauge  to.  Then  a  line  on  the  center  of 
the  top  of  the  rafter  in  connection  with  a  line 
gauged  on  the  side,  will  give  the  bevel  or  back- 
ing. 

Another  example  of  backing  and  I  have  done. 
Let  us  suppose  AB  and  BC,  Fig.  78,  to  represent 
the  plates  of  the  building,  and  BD  the  hip  rafter. 


THE  STEEL  SQUARE 


"7 


BE  being  the  seat  of  the  rafter.    Take  any  point 
of  the  hip,  as  i.     Draw  a  line  at  right  angles  to 

0 


Fig.  78 
this,  producing  it  until  it  cuts  seat  BE,  as  shown 
in  the  point  2.  From  the  po!  t  2  thus  estab- 
lished draw  a  line  perpei  licular  to  the  seat,  pro- 
ducing it  until  it  cut  ;  line  of  plate  AB. 
Transfer  the  distance  1  along  the  line  repre- 
senting the  seat  of  the  rafter,  thus  establishing 
the  point  4.  Draw  3  and  4;  then  at  4  ••11  be 
given  the  bevel  for  use  in  backing  the  rafter. 
Fig.  79  shows  the  application  of  the  bevel  to  the 
timber  which  will  give  the  gauge  points  to  work 
from. 


'■    li 


■-.   •-•t 


•     t 


I  H 


ii8 


PRACTICAL  USES  OF 


These  examples,  with  the  ones  on  the  same 
•bject  illustrated  and  described  in  previous 
pages,  shoulw  prove  quite  am- 
ple and  are  varied  enough  to 
meet  the  requirements  of 
most  workmen  no  matter 
what  may  be  their  constitu- 
tional peculiarities. 

Fig.  79 

FRAMING    OCTAGONAL   ROOFS,    DOMES,    BAYS    AND 
OTHER    OCTAGONAL    WORK 

We  now  enter  another  phase  of  the  carpen- 
ter's art,  and  one  in  which  the  steel  square  plays, 
or  can  be  made  to  play,  an  important  part.  I 
have  discussed  the  "octagon"  pretty  fairly  in  the 
first  volume  of  this  work,  but  very  much  more 
than  I  have  said,  or  can  say  for  that  matter,  may 
be  said  on  the  construction  of  octagonal  work; 
in  order,  however,  to  make  this  work  as  com- 
plete as  possible  I  have  thought  it  necessary  to 
present  to  the  reader  the  following  illustrations 
and  descriptions,  knowing  from  experience  they 
will  be  useful. 

I  have  shown  how  the  miters  in  polygons  may 
be  obtained  by  aid  of  the  square  and  by  other 
methods,  and,  as  a  sort  of  introduction  to  this 
chapter,  I  offer  the  following  which  I  know  will 


THE  STEEL  SQUARE 


119 


\ 


be  acceptable  to  many  of  the  readers  of  this  book 
who  have  been  fortunate  enough  to  get  a  fair 
public  school  education:  There  are  three  kinds 
of  angles:  the  right,  the  obtuse,  and  the  acute. 
A  right  angle  is  an  angle  formed  by  two  lines 
perpendicular  to  each  other.  An  obtuse  angle 
is  greater  than  a  right  angle;  an  acute  angle  is 
less  than  a  right  angle.  All  angles  of  the  octa- 
gon are  obtuse.     A  right  angle  is  equal  to  90°. 

The    angle    ABC,  _r - >. 

Fig.  80,  which  is 
one  of  the  angles 
of  the  octagon,  is 
45'  greater  than  a 
right  angle,  and  is 
equal  to  90°+45°=  osi'" 
135'.  The  octagon 
miter  is  an  acute 
angle,  and  is  found 
by   bisecting   135°,  Fig.  80 

which  is  'i''=67~  30',  which  is  shown  at  ABD. 
I  will  now  show  the  proportionate  length  of 
each  line  in  the  octagon,  Pig.  81;  the  diameter 
being  one,  the  number  on  each  line  indicates  its 
exact  length  in  fractional  parts  of  one.  To  lay 
out  the  miter  or  angle,  place  the  square  as  shown 
at  ABC;    take   12  inches  on  the  blade  of  the 


} 


m 


u^ 


, 

i 

t  ■ 

1 

lao 


PRACTICAL  USES  OF 


square  and  4^  on  the  tongue;  tongue  gives  cut. 

Any  other  number  will  do  as  well,  providing  the 

"l  proportion  of  3827 
and  1585  exists  be- 
tween them. 

This  is  a  simple 
and  correct  meth- 
od  of   finding  the 
miter   of  an   octa- 
gon,  and    will    be 
found    useful    in 
solving  many  prob- 
lems that  confront 
the  workman  from 
time  to  time. 


Pi'-.  81 


BAV   WINDOWS 

Often  workmen  are  put  to  their  "wit's  end" 
when  "laying  out"  an  octagon  bay  window,  owing 
to  the  surrounding  conditions.     The  following  is 
submitted,  which  shows  how  the  faces  or  sides 
of  the  wmdow  or  other  w^ork  may  be  laid  out 
with  ease:     First  lay  off  a  straight  line  DA.  l-jg. 
^2,  to  the  length  desired  for  one  side  of  the  win- 
dow, as  indicated  from  A  to  B.     Then  from  B  to 
C  make  the  length   A  of  AB.     The  length  CD 
IS  to  be  the  same  as  AB.     Now,  with  the  foot  of 


THE  STEEL   SQUARE 


lai 


the  compasses  in  D,  and  with  radius  DC,  strike 
an  arc  as  shown.     Then,  with  the  same  radius 

•  E -^   r. 


Fig.  32 

from  A  as  center,  strike  the  second  arc  indicated. 
With  the  dividers  set  to  the  same  distances  and 
with  C  as  center,  strike  an  arc,  cutting  the  arc 
^iruck  from  A,  thus  establishing  the  point  F; 
then,  in  the  same  way,  using  B  as  center,  strike 
an  arc  cutting  the  opposite  arc,  establishing  the 
point  E.  Draw  the  lines  DE,  EF,  and  FA,  the 
result  will  be  three  equal  lengths  and  three  equal 
angles.  To  find  the  center  of  the  octagon,  draw 
lines  through  the  points  FB  and  EC  until  they 
intersect  in  the  point  G;  then  G  will  be  the  cen- 
ter as  required.  The  lines  FB  and  EC  will  be 
the  seats  of  hips,  if  any  are  desired.     To  lay  off 


i"' 


if 


il;  .* 


laa 


PRACTICAL  USES  OP 


.T^'t 


1 


ii 


I  i 


an  octagon  end  of  building,  as  is  often  done, 
divide  the  width  of  the  building  into  29  parts, 
and  take  12  parts  for  each  of  the  extreme  spaces 
and  5  parts  for  the  mean  space,  and  proceed  as 
above.     If  we  wish  to  make  the  front  side  wider 
than  the  other  side— for  example,  2  feet  wider— 
deduct  2  feet  from  the  width  of  the  building; 
divide  the  remaining  space  into  29  parts,  take  12 
parts  each  for  the  extremes  and  5  parts  plus  2 
feet  cut  off  for  the  mean  space,  and  proceed  as 
above,  save  that  in  crossing  the  arc  at  E  wt  must 
set  the  compasses  5  parts  from  C.  or  at  I,  all  as 
shown  in  Fig.  83.     And  in  crossing  at  F  we  set  5 

parts  from  B,  or  in 
the  point   H,  a*- 
shown.      Then  we 
have  the  front  side 
2  feet  longe-  than 
the  others,  and  the 
angles    the    same. 
Three  sides  of  any 
figure  composed  of  more  than  four  sides  can  be 
produced   in  the  same  general   manner.     How- 
ever, the  ratio  between  tho  mean  part  and  the 
extremes  will  be  different.     Thus,  in  a  figure  of 
seven  sides  the  mean  part  will  be  one-fou'  h  of 
the  extremes.     Whatever  the  mean  part  is,  the 


Fig.  83 


:i    ) 


'   l-i^i  - 


THE  STEEL  S(^UARE 


i»3 


.ides  will  be  equal  and  the  angles  at  E  and  F 
will  be  the  same. 


Fig.  84 
OCTAGON    TOWKKS    .\Nn    SPIRES 

Now  iiat  we  know  how  to  lay  out  the  base  of 
an  octagon  and  how  to  lay  off  a  part  of  the  fijrure 
for  a  bay  windc  w  or  other  similar  work,  it  will  be 
in  order  to  see  how  the  framing  is  done  for  an 


\  m 


194 


PRACTICAL  USES  OP 


octatronal  tower,  spire,  or  other  simihr  structure. 
Suppose  we  have  a  tower  to  erect  which  is  partly 
over  another  roof,  as  shown  at  Fig.  84,  where 
the  intersections  occur.  It  will  be  seen  that  the 
tower  intersects  the  hip  roof,  as  A,B,C,D,E,F,G. 


Fig.  85 


Before  the  intersections  shown  by  Figs.  84  and 
85,  and  the  timbers  shown  by  Figs.  87,  88.  and  89 


lit 


THE  STEEL  SQUARE 


»«1 


*l 


i-'    i\ 


!  :    t! 


I  '    I 


'       I 


Fig.  86 


mLM 


hu 


^  m^:^*^>  '^^g^Jil^^^, 


Fig.  87 


can  be  properly  set  out,  it  will  be  necessary  to 


ii_ 


mmmi^M^^3t^^'^^ii^J(fi--^.^^M-  .5N&^..:^-m'x:'::^mri 


>< 


THE  STEEL  SQUARE  ,,7 

obtain  the  Intersections  of  the  boarded  surfaces 
{^n^ometri-ar-y.  The  method  of  doing  this  is 
shown  ,>y  Fig.  r.r,,  .n.d  is  as  follows:  Set  out  the 
half  oc  ayon  A^iS  B,  which  is  the  line  of  board- 


FiG.  88 

ing.  The  other  half,  A3456B,  it  will  be  noticed, 
IS  a  little  less,  this  being  the  line  of  rafters.  To 
avoid  confusing  the  diagrams  with  a  number  of 
lines,  several  of  them  have  been  omitted;  it 
would  of  course  form  a  smaller  parallel  octagon 
to  those  shown.  Next  set  out  line  CD.  which  is 
the  line  of  feet  of  rafters,  and  P:F,  which  is  the 
line  of  face  of  the  fascia  board  of  the  main  roof, 
also  the  line  of  the  main  hips,  as  shown  at  GH. 


hi 


ij 
!i1 


138 


PRACTICAL  USES  OP 


At  right  angles  to  i-8  draw  OP,  and  at  right 
angles  to  this  line  set  up  OR,  making  it  equal  to 
the  height.  Join  PR,  which  is  the  true  inclina- 
tion of  the  sides  of  the  lower  roof.  At  any  point 
along  EF  draw  xy  at  right  angles  to  it,  and  set 
up  the  pitch  of  the  main  roof  as  shown  at  xs. 


Fig.  8g 

Now  take  any  point  T  on  this  pitch  line,  and 
project  down  at  right  angles  to  xy,  meeting  it  as 
shown.  From  O  mark  off  OV  equal  to  the 
height   TU.      From    V   project    across    to    W, 


.^^ff^fJIMi^3V»mm*L«_. 


THE  STEEL  SQUARE 


139 


parallel  to  1-8,  which  will  meet  TU  in  a.  Join 
Ea,  which  will  give  the  intersection  of  the  sur- 
face 0-1-8,  and  the  main  roof.  For  the  next  inter- 
section, from  where  Wa  cuts  0-8  in  ^,  draw  a  line 
parallel  to  7-8.  Now  produce  TU,  which  meets 
the  last  line  in/.  Then  from  d  draw  through/ 
to  meet  0-7  in  ^.  This  gives  the  intersection  of 
the  main  roof  with  the  triangular  portion,  0-7-8. 
The  side  B7  should  be  continued  so  as  to  meet 
EF  in  //.  Join  %  and  produce  to  G.  Then  ^G  is 
half  the  intersection  of  the  surface  7-0-6.  Work- 
men having  a  knowledge  of  geometry  will  see 
that  the  principle  of  working  this  has  been  based 
on  a  problem  in  horizontal  projection,  the  spe- 
cific p— blem  being.  "Given  the  horizontal  traces 
and  :linations  of  planes,  find  their  intersec- 

tions.' 

If  it  is  desired  to  obtain  the  developments  of 
the  several  surfaces,  they  can  be  obtained  in  the 
following  manner:  Draw  PO  at  right  angles  to 
1-8,  and  OR  at  right  angles  to  OP.  Measure  in 
OR  the  height.  Join  PR,  which  gives  the  incli- 
nation and  true  length  of  the  center  line  of  the 
full  surfaces.  Bisect  line  2-3,  and  at  right  angles 
to  it  draw  AK,  making  AK  the  same  length  as 
PR.  Join  2K  and  3K,  which  gives  the  true  shape 
of  each  of  the  full  surfaces.    This  development 


lki*i^9mWiWA% 


m 


,V'.. 


133 


PRACTICAL  USES  OF 


can  be  used  to  shr  vv  the  correct  shape  of  the 

surfaces    which    intersect    the    roof.       From   i 

project  up  to  meet  tl.o  line  PR  in  //;    make  AZ 

equal  to   Fd';    then  through   Z   draw  line  ,i^7.M 

parallel    to   A3;    next    make   3L   equal    to    E8. 

Then  2LMK  is  the  true  shape  of  the  side  IFJO. 

From  ^  draw  a  line  parallel  to  7-8,  meeting  8-0  in 

K;  then  from  the  point   K  draw  a  line  parallel 

to  1-8,  and  continue  it  to  meet  PR,  and  meeting  />. 

Now  measure  off  on  the  line  AE  a  distance  AX 

equal  to  P/.     Through  X  draw  a  parallel  to  2-3, 

meeting  3K  in  point  X.     Join  N^/;   then  Ny/-  is 

the  true  shape  of  the  surface  _ir/)o.     From  G  draw 

GwHS  parallel  to  B78P.     Trom  s  project  up  to 

meet  PR  in  r.     Make  A;/  equal  to  rP.     Join  X^^ 

and  nv;   then  X//<'K  is  the  true  development  of 

the  surface  JG^O. 

The  method  of  obtaining  the  bevels  of  the 
several  parts  may  now  be  described,  the  meatr-- 
of  obtaining  the  backing  of  the  hips  br^ing  fir'-t 
shown.  At  right  angles  to  0-4  set  0-9,  and  make 
it  equal  to  the  hef'^ht;  join  4-9,  which  gives  the 
true  rake  of  the  hips.  At  11  is  shown  the  bevel 
for  the  vertical  cut  of  the  top,  and  at  12  that  for 
the  foot.  As  will  be  seen,  one  edge  of  this  bevel 
is  adjacent  to*  the  pitch  line;  the  other,  being 
horizontal,   is  drawn  parallel  to  0-4.      For  the 


*ll^'-  :^vm^j!^^9m'tj^ 


THE  STEEL  SQUARE 


131 


backin{(  of  the  hips,  join  3-5,  and  from  where  this 
line  meets  0-4  in  point  10,  draw  an  arc  tangent  to 
the  pitch  hne  4-9.  From  where  the  arc  meets 
4-10  in  point  11,  join  to  5  and  3  as  shown;  then  D 
is  the  bevel  required. 

The  bevel  for  where  the  hips  meet  each  other 
is  shown  at  13.     Reference  to  Fig.  8}  will  show 
where  this  bevel  will  be  required,  and  also  that 
the  upper  part  of  the  mast  or  central   post   is 
octagonal.     This  allows  the  upper  cuts  of  the 
hips  to  be  made  square  through  their  thickness, 
and  therefore  no  bevel  is  required.   The  develop- 
ment of    the  intersection  shown  at  3MXKe'^2 
gives  us  the  bevels  for  the  feet  of  the  hips  and 
rafters  :A,  B,  C,  D,  E,  F,  and  G,  Fig.  87.     The 
bevels  14,  15,  16,  and  17,  Fig.  86,  are  the  feet  of 
the  hips  A,  B,  E,  and  F,  respectively,  Fig.  87. 
These  bevels  are  for  application  after  the  hips 
have  been  backed.     The  bevels  to  apply  to  the 
backs  of  the  jacks  at  C  and  D,  Fig.  87,  are  shown 
at  18,  Fig.  86. 

Whilst  the  bevel  for  the  foot  of  G,  Fig.  87,  is 
shown  at  19,  Fig.  86,  it  will  be  noticed  that  the 
valley  rafters  shown  by  i,  2,  3,  Fig.  87,  have  their 
upper  edges  in  the  same  plane  as  the  main  roof; 
therefore,  it  will  be  necessary  to  obtain  a  bevel 
for  the  prej^aration  of  these  edges.    The  geo- 


Jl 


M 


^?&l^.V^.m*^-  :W. 


'3* 


PRACTICAL  USES  OP 


metrical  construction  for  this  is  as  follows:  From 
any  point  in  the  plan  of  the  valley,  as  H,  Fig.  87, 
draw  the  horizontal  lines  HK  and  HL  at  right 
angles  to  2-3;  then  at  any  point  on  HL  draw  X  Y 
at  fight  angles  to  it,  and  cutting  the  line  HK  in 
K.     From  where  HK  cuts  the  pitch  of  the  roof, 
as  shown  at  M,  draw  MX  at  right  angles  to  HK. 
Then  from  L  drop  a  perpendicular  to  MX  as 
shown.     Xext  project  Li*  at  right  angles  to  XY, 
and  make  it  equal  in  length  to  XO.     Xow  join 
KP;    then  with   L  as  a  center,  draw  an  arc  tan- 
gent to  KP,  meeting  XY  in  R.    Join  RH,  which 
gives  the  bevel  required  as  shown  at  25. 

The  bevels  for  the  jack  are  shown  at  20  and  21, 
Fig.  86,  whilst  the  bevel  at  13  is  for  application 
to  the  tops  of  the  jack  rafter,  or  these  bevels 
may  be  obtained  by  the  steel  square  as  shown  in 
previous  examples.  The  methods  for  obtaining 
the  bevel  for  the  jack  rafters  for  the  main  roof 
may  be  obtained  by  the  square,  as  shown  by  22. 
23,  and  24,  Fig.  86,  respectively.  The  hip  of  the 
main  roof,  as  will  be  noticed,  requires  supporting 
at  the  lower  end.  This  is  done  by  placing  a 
dragon  beam  across  the  octagonal  space,  as 
shown  at  S  and  S,  Figs.  87  and  88.  Then  the  end 
of  the  hip  should  be  notched,  as  shown  at  Fig. 
go.      The  ceiling  joists  in  the  octagonal  space 


..^H^f 


.  'w : '--' 


THE  STEEL  SQUARE  .  ,33 

are  fastened  into  the  wall  as  shown  on  plan, 
Figs.  87  and  88.     Of  course,  as  is  usual,  -he  ceil- 


Fio.  90 

ing  joists  should  be  on  the  same  plane  as  the 
ceiling  joists  in  the  main  building;  the  ends  of 
four  of  these,  U,  V,  W,  and  Z,  cannot  be  carried 
to  any  wall,  therefore  a  trimmer  is  provided  of 
stouter  scantling  to  carry  these  ends,  as  shown 
in  the  plan  and  section,  Figs.  87  and  88.  The 
boarding  is  clearly  shown  in  Figs.  84  and  85,  and 
therefore  does  not  require  further  description. 
There  are  other  little  points  which  are  fully 
shown  in  the  illustrations,  but  it  has  not  been 
thought  necessary  to  enlarge  upon  them  here. 


m 


.4%^  '  %JM 


:! 


1  i 


•  1 


'34 


PRACTICAL  USES  OF 


At  Figs.  88  and  89  the  man- 
ner of  construction  is  shown, 
including  projeciion  of  raft- 
ers over  eaves.  It  will  be 
noticed  there  is  a  center 
post  to  which  the  hips  or 
corner  rafters  are  nailed. 
This  post  is  not  absolutely 
necessary,  but  when  it  can 
be  used  it  is  a  great  help  to 
rapid  construction,  and  cer- 
tainly makes  the  work 
stronger. 

The  pitch  of  a  tower  roof 
may  be  obtained  along  with 
all    the    bevel     lines    by    a 
proper  use  of  the  square,  as 
shown  in  P'ig.gi,  which  'llus- 
trates  some  unusual  pitches. 
It  is  evident  that  if  the  run 
of  one  foot  is  12  inches  the 
run    of    two    feet    must   be 
double    that   or   24    inches. 
Therefore  the  rise  must  be 
that  proportion  of  24  inches. 
The  first  inch  in  rise  is  sV,    CZH] 
the  second  ,'2,  and  the  third 


t*ftt-i 


-SAtm 


■i/ Pitch 


-2PiTeii 


-li  Pitch 


iSj-I  Pitch 


Fig.  91 


lis 


^j^..\^. 


THE  STEEL  S(JUARE  ,35 

14,  the  fourth  /t,,  etc.  The  twenty-fourth  inch 
rise  being  equal,  the  span  is  therefore  i  pitch. 
As  the  rise  continues  above  this  point,  it  is  simply 
a  repetition  of  the  above  with  a  i  prefixed,  thus: 
Th-  tvventy.fifth  inch  rise  being  a  pitch',  etc.; 
but  we  are  now  beyond  the  limits  of  the  full 
scale  as  applied  to  the  square,  so  we  must  reduce 
the  scale. 

By  letting  the  vertical  line  ai  A  represent  the 
blade  we  will  have  reduced  th(-  scale  one-half. 
The  pitches  would  center  at  6  on  the  tongue 
instead  of  12.  as  in  the  full  scale.  We  must  now 
use  the  half  inches  above  12  on  the  blade  for 
each  inch  in  rise  till  we  reach  the  twenty-fourth 
inch  which  will  be  equal  to  2  pitches  or  48-inch 
rise  to  the  foot. 

For  steeper  pitches  it  is  necessary  to  again 
chan  -  the  scale.  •  If  we  let  the  blade  rest  at  B 
the  patches  will  center  at  3  on  the  tongue  (ma- 
king the  scale  %  size),  and  by  letting  the  K  inches 
abo/e  12  on  the  blade  represent  the  full  inches 
in  rise  will  give  the  cuts,  etc..  from  the  forty- 
eighth  inch  rise  to  the  ninety-sixth  inch  rise  to 
the  foot,  or  4  pitches. 

One  of  the  methods  of  laying  off  the  lines  for 
an  octagonal  roof  having  curved  rafters  is  given 
at  Fig.  92,  where  a    method  of  obtaining   the 


!  i   - 

1   'f 


lli 


f 


i'  r 


136 


PRACTICAL  USES  (JF 


.:urves  Is  srlvcn.    The  plan  is  shown  by  the  octag- 
onal figure,  and  we  will  suppose  it  to  be  20  feet 


-8  4 


Fic 


in  d 


iameter  and  the  rise  of  roof  25  feet. 


ABis 


THE  STEEL  SQUARE  ,.„ 

the  run  of  t^c  common  rafter  and  AC  its  rise. 
Divide  AB  into  as  many  parts  as  may  be  neces- 
sary and  square  up  from  each  of  these  points 
parallel  to  AC,  and  cutting  the  curved  line  CB, 
which  in  this  case  is  struck  with  a  sweep  of  38 
feet.     Now  divide  the  run  of  the  rafter  AD  into 
the  same  number '  f  equal  parts  as  the  run  of  the 
common  rafter  A'.i.     P:rect  perpendiculars  from 
each  of  these  points  at  rifrht  angles  to  AD  and 
set  off  from  A  to  E  the  same  distance  as  that 
from  A  to  C  on  the  common  ratter.     Next  set 
off  II,  22,  33,  etc.,  on  the  hip  to  correspond  with 
II,  22,  1%  etc.,  on  the  common  rafter,  and  con- 
nect these  points.     The  result  is  the  shape  of  the 
hip  rafter.     As  for  the  jack  rafters,  their  lengths 
depend   on   the  number   necessary.      If  one   is 
sufficient,  its  length  would  be  one-half  that  of 
the  common  rafter  taken  on  the  working  line, 
and  hy  making  the  cut  for  the  upper  end  through 
this  point,  as  shown   at  4  where  we  have  the 
proper  length  of  the  jack.     The  plumb  and  ver- 
tical cuts  of  the  jacks  are  the  same  as  those  on 
the  common   rafter.     The   figures  on   the  steel 
square  vvhich  ^ive  the  cuts  for  an  octagon  are  7 
inches  and  17  inches;    the  7-inch  side  giving  the 
cuts.     The  figures  on  the  square  which  will  give 
the  cuts  for  common  rafter  in  this  case  are  5 


I         I 


: 


Tl" 


^ 


»38 


PRACTICAL  USES  OP 


inches  on   the  tongu..  and    ,2^   inches  on   the 
blade:    the  latter  giving  the  upper  cut  and  the 
tongue  the  lower  cut.     The  figures  giving  the 
cuts  for  the  hip  rafter  in  this  case  are  5  and  5^ 
•nches  on  the  tongue  and   ,2>^  inches  on  the 
blade;    the  latter  giving  the  plumb  cut.     This 
will  he  readily  understood  for  the  reason  that 
the  run  of  a  hip  rafter  on  an  octagon  is  one- 
twelfth  greater  than   the   run   of  the  common 
ra  ter.     One  method  of  obtaining  cuts  of  the  jac  k 
ratters  is  shown  on  the  lines  Ati  and  liC      It  will 
be  seen  that  it  will  work  in  any  case,  no  matter 
what  may  be  the  pitch  of  the  roof  or  the  shape 
of  the  rafter.     Obtain  the    plumb  cut  of    the 
upper  end  of  the  rafter  EC.  which  is  the  same 
as   that  of  the  common   rafter.      Then    square 
across  from  A  to  H  on  the  upper  edge.     Now. 
as  7  inches  and  ,7  inches  on  the  square  will  give 
the  cuts  of  the  jacks  if  they  are  to  have  no  rise 
at  all,  the  same  will  work  when  they  have  a  rise 
I  ake  seventeen-seventiis  of  the  thickness  of  the 
stuff  whjch  is  being  worked  and  set  it  off  square 
from   the  line  BC  to  the  outer  edge,  as    CD. 
Then  a  hne  fn.m  A  to  C  is  the  bevel  c,f  the 
jacks.      All    these    cuts    and    lengths    may    be 
obtained  by  using  the  square,  as  has  been  shown 
in  previou'i  examples. 


THE   STEEL   SoUARE 


«39 


Another  method  of  obtaining  the  curves  for 
hip  and  jack  rafters  is  shown  at  Figs.  93  and  94. 

The    lengths    and     bevels    will,    of 

( ourse,  be  the  same  as   though  they 

were  to  be  straight,  and 

lay  them  out  in  that  way  tA/T" 

on    boards  wide  enough 
i  make  the  curve.     It  is 

^est    to    ha  v  e    the  m 

planned   and  jointed  on 

the  back,  then  strike  the 
curve    of    the    common 
rafter.     It  may  be  struck 
with  a  trammel  from  one 
renter,  as  in  ihe  sketch, 
or  of  any  shape  that  may  i 
suit  the  fancy  or  condi-? 
tions  of  the  case.     Next.i 
divide  the  length  of  the' 
rafter  on  the  jointed  edge 
of  the   pattern  into  any 
number   of  equal    parts 
and  draw  the  lines,  as  i, 
2.  3,  etc.,  in  the  sketch, 
Fig.  94,  on  the  same  bevel 
as  the  plumb  cut.     Then 
proceed   in    the   same 


'  "^-  93 


i.Si. 


I40 


«  I 


i  f| 


PRACTICAL  USES  OF 


manner  wfth  the  board  for  the  hip  rafter,  being 
careful   to  divide  it  into   the  same  number  of 


i 


"j^mik 


THE  STEEL  SQ^JARE  ,4, 

equal   parts,   and  draw  the  lines  parallel   with 
•ts  plumb  cuts.     It  will  be  found  convenient  to 
number  the  lines  on  the  patterns  the  same  as 
shown  m  the  sketch;   then,  with  the  dividers  or 
ruj^e,  lay  off  i'  on  the  hip  equal  to  i  on  the  jack. 
2     2,  3  -3,  etc.    Then  spring  a  light  line  or  edg- 
ing  and  draw  through  the  points  thus  obtained 
If  the  work  is  done  correctly  the  two  sides  of  the 
roof  will  meet  exactly  on  that  line.     It  will  be 
readily  seen  that  it  makes  no  difference  whether 
the  hip  IS  to  be  set  on  a  square,  hexagon,  octa- 
gon, or  at  the  angle  of  any  other  regular  figure 
providing  run  and  length  are  first  properly  set 
off.    Another  method,  which  may  be  termed  a 
geometrical  method."  is  shown  at  Fig.  95  and 
to  those  who  have  any  knowledge  of  geometry 
further  explanation  will  be  unnecessary.     It  is 
given  here  merely  as   a  comparison  and  may 
perhaps,  be  found  useful  to  a  few  readers. 

The  plan  and  elevation  shown  at  Fig  96  is 
almost  self-explanatory.  It  simply  shows  the 
lengths  and  bevels  of  hips  and  jacks.  The  cuts 
>n  both  h.ps  and  jacks  are  the  same  as  would  be 
tor  common  rafters,  except  thut  instead  of  a 
square  cut  across  the  back  of  the  rafter  it  must 
he  at  a  diagonal  to  fit  against  the  side  of  the  hip 
as  shown  by  the  dotted  lines  at  A  and  B 


w 


!,     8 


M* 


PRACTICAL  USES  OP 


:11 


!  i 


i 


I 
I 


These  lines  are  always 
vertical  and  the  same  dis- 
tance apart  regardless  of 
the  pitch  given.  A  diag- 
onal line  from  A  to  B 
across  the  back  of  the  jack 
determines  the  angle. 

Fig.  97  illustrates  this 
point.     If  there  was  not 
pitch  at  all  then  5  and  12 
would  give  the  cut.  These 
figures  also  give  the  start- 
ing points  of  lines  A  and 
B,  which,  since  the  rafters 
are  of  the  same  thickness, 
will  remain  at  right  angles 
the  same  distance  apart. 
Thus,  if  the   rafter  be  2 
inchesthick.thelinesAand  • 
B  will  be  45i  inches  apart. 
The  jack  cut  may  also 
be  found  as  follows:  Take 
5  on  the  tongue  and  the 
length    of    the    common 
rafter   for  one   foot   run 
on  the  blade,    the  blade 
giving  the  cut. 


.  '!*£ ,  'T I  «*  ivKT  '*  "klV',  /  KJy  vA 


"HE  STEEL  SQUARE 


•*i 


Ij 


^  a»^»f■^^v 


Fn:.  97 

I'i^.  98  is  a  modified  diagram  of  Fig.  96. 

Ot:TAGO\    DORMERS 

I  have  thought  fit  to  present  to  my  readers  a 

few  .llustrafons  showing  how  an  octagonal 
dorn,      „  ,„  ^^^  ^^^^.^^  ^^  ^  ^^.^^^^ 

roof  by  architects  of  note.  But  little  more 
s  necessary  than  the  plans,  sections,  and  eleva- 
■ons  showing  the  mode  of  construction  to  enable 

.he^rcader   to    understand  the  whole  arrange- 

IV  99  shows,  in  a  conventional  w.j,  how  the 

"be  s  are  arranged  and  fr,m«l  in  order  to 

■n^l^e  the  dormer  bay  strong  and  effective.    The 


"M 


■  n 


wjtri.^?sa^z 


<  i 


f" 


!!«F 


r 


n 


»*4 


PRACTICAL  USES   JP 


sidelights  in  a  dormer 
of  this  kind  are  gener- 
ally fixtures,  while  the 
front  sashes  may  be 
hinged  and  open  in  two 
leaves,  or  they  may  be 
so  constructed  as  to 
hang  with  cords  and 
weights. 

Fig.  ICO  shows  a  front 
elevation  of  the  frame- 
work with  sashes  in 
place;  and  fall-back  ga- 
ble of  roof,  rafters,  and 
all  studding  are  seen 
in  position.  Fig.  loi 
shows  a  plan  of  the 
whole  construction,  in- 
cluding top  of  brick 
wall,  plate,  and  gutter. 
This  figure  requires  no 
further  explanation. 

Figs.  I02  and  103  show 
side  elevations  of  por- 
tions of  roof  and  dor-^ 
mer. 


Fig.  98 


1HE  STEEL  SQUARE 


»45 


Pig.  99 


[:|lf| 


i^l 


i  ,    ! 

•  ^   t! 


r-Ki 


.■>«rrT'*B  '.'<«3R1^  ^' 


tkr  - "  ■  ««».-«fc:>ifi-    Tt:     vOl 


146 


:  i  1 


I     i 


PRACTICAL  USES  OP 


At    Fi^r.    ,04  a   ground   plan   of    the  dormer 
proper  is    shown   giving  shape  of  corner  and 


Fic.  100 


angle  posts.  A  little  study  of  these  examples 
will  enable  the  reader  to  understand  the  prin- 
ciple of  construction  without  further  explana- 
tions. The  steel  square  may  be  used  to  get 
every  cut  and  bevel   in  this  roof,  also  bevels  for 


SWCT  &«*&;« 


Xi.'SBKKr  "¥8 


THE  STEEL  SQUARE 


147 


the  window  sills,  as  shown  in  Fig.  105.  and  which 
IS  explained  as  follows:      Take  the   height   of 


Fio.  loi 
rise  A  to  A'  and  set  up  from  B  to  C  each  way 
Intersect  the  two  lines  at  D.one  at   .n  arc  struck 
troni  O  as  a  center,  and  the  other  from  E.     The 
line  from  O  to  D  will  be  the  bevel  for  face  of 


:j|j 


m 


148 


PRACTICAL  USES  OP 


if ! ' 


Fta.  :o3 


°A^. 


THE  STEJiL  SQUARE  ,4, 

sill.    To  obtain  the  down  cuts,  drop  a  line  from 
the  point  of  overhang  of  sill  to  the  line  of  inter- 


Pio.  104 


section  of  the  angle  of  the  bay  window.    Set  of! 
the  thickness  of  sill  parallel  to  face  of  window. 


*   t«iht 


Fig.  105 

Square  up  from  where  the  point  of  overhang  cuts 
the  Ime  of  intersection  of  angle  to  the  thickness 
of  sill  and  draw  a  line  from  H  to  O.  which  will 
be  the  bevel  or  down  cut.  This,  of  course,  gives 
he  bevels  for  all  sides  of  the  sills.  At  Fig.  106 
i  show  an  octagon  tower  in  place,  or  rather  the 


!l 


•H 


m: 


-i 


i 


I': 


ir  ri 


11 


li'^n 


ISO 


PRACTICAL  USES  OP 


method  of  framing  same  on  a  balloon  building 
over  a  veranda.  Sizes  of  timbers  are  shown. 
The  example  may  be  of  use. 

Fig.  107  shows  a 
plain  octagonal 
tower  finished. 
This  is  in  connec- 
tion with  a  two- 
storied  frame 
house  that  is  sided 
outside.  A  por- 
tion of  veranda 
and  front  entrance 
i  s  shown.  The 
whole  is  made  as 
plain  and  econom- 
f  al  as  possible. 
.Vhile  these  last 
two  figures  have 
no  direct  connec- 
tion with  the  steel 
square,  it  is 
thought  they  may 


a'xfl' 


o  xe 

Fig.  106 

e  useful  as  showing  the  work  when  finished. 


^^,-yMksM^ 


^nkJ^Si-  u-^lt^J 


THE  STEEL  SQUAKE 


«S« 


f   -i 


ijiiiii»w»iii»wiiiMiiii»»»i»iiiiiiihiiuiiiii(.wti»ywiiiiia'yt'g<*y'i!*''' 
Fiu,  107 


if 


<      ?! 


..£1 


j^   '.'. 


I, 

i  I 


I  i 


•    a!  ^  • 


i 


>i«  PRACTICAL  USES  OF 

liOPPERS  AND  HOPPER  BEVELS 

•he   question   of  "hopper  cuts'    is  one   that 
•  CMS    o  puzzle  nearly  all  youi  -  workm-  n-and 
n.any     -Id      n-s    also.      Inde-xi.   I    have    known 
many  ixctllenr  workmen   who  coi  Id  cut  every 
lin'.Lx  ■      or    a    complicated    hij)    roof,    on    the 
►-oun',   I, HJ  whose  work  was  beyond  suspicion, 
who  couid  not  lay  out  the  lines  for  the  miter 
cuts  of  a  hopper  in  a  proper  manner.     In  this 
.  hapter  I  will  endeavor  to  ^dve  to  the  reader,  in 
as  simple  manner  as  I  know  how,  a  number  of 
the  best  methods  employed  by  expert  workmen 
for  findinjr  the  proper  lines  for  cuttin^^  hopper 
bevels,  both  by  geometrical  methods  and  by  the 
use  of  the  steel  square,  and  to  this  end  I   have 
gathered  up  a   number  of   methods,  diagrams. 

and  explanations  .rom 
various  sour  es — many 
of  which  have  heen 
published  before. 

The  shapes   of   hop- 
pers shown  at  Figs.  108, 
log,  and    no  nre  trian- 
gular, square,  and  hf  v. 
agonal  respectively,  while  Fig.  ,1,  shows  a  cover 


Fig.  108 


THE  STEEL  SQUARE  153 

or  box  lid,  which  requires  a  little  special  treatment. 


Fir,     109 

I  will  treat  all  these  figures  in  u  geometrical     ay, 

so  that  the  reader  may  know 
the  "reason  why"  It  is  nee  -*- 
sary  for  C(  -  in  cuts  to  ha  e 
certain  lin^s  ihaf  are  at  va- 
riance  vith  cer      n  lines  in 


Fio.  no 

apparently-  simi- 
lar cables.  The 
j^  e  o  m  » •  t  r  i  c  a  1 
P  r  o  b  I  e  m  s  i  n- 
volved  in  these 
cases  consist  (1)  in  ti   di    4 


Fii,,  III 

dihedral  angle  be- 


liji 


■4 


154 


PRACTICAL  USES  OF 


■  ;! 


tween  two  planes;  (2)  in  bisecting  this  angle,  which 
gives  the  bevel;  (3)  in  developing  or  obtaining 
the  true  shape  of  a  plane  surface.  A  geometrical 
problem  of  a  very  ordinary  type,  involving  the  first 


VlG.    .H 

two  of  this  character,  is  shown  by  Fig.  11 2,  where 
two  planes  are  indicated  by  their  horizontal  and 
vertical  traces.  The  solution  is  as  follows:  First 
find  the  plane  and  elevation  of  the  intersection 
odAE.  Next  obtain  the  true  length  of  the  inter- 
section by  rebating  it  into  the  horizontal  plane  as 
shown  by  ^B.  Through  any  point  c  on  a  draw  de 
at  right  angles  to  ab.  Then  from  point  c,  draw  cf 
at  right  angles  to  «B.  Then  with  c  as  center  and 
/  as  radius  cut  ab  in  g.  Join  eg^  and  dg;  then  dge  is 
the  dihedral  angle  between  the  two  planes. 
Then,  given  two  pieces  of  material  of  equal  thick- 


.P*C«^^i^T?iff 


THE  STEEL  SQUARE 


^S5 


ness  meeting,  and  it  is  desired  to  miter-joint 
them,  the  angle  for  this  would  obviously  be  half 
the  dihedral  angle,  as  A. 

Let  this  geometrical  reasoning  be  applied  to 
the  case  illustrated  by  Fig.  113,  which,  as  will  be 
seen,  is  a  direct  application  of  the  problem  just 
described,  ad  indicating  the  plane  and  a'6  the 
elevation  of  the  line  of  intersection  of  the  two 
adjacent  plane  surfaces.  Draw  6"  at  right  angles 
to  ad,  and  make  it  equal  to  M;  then  joining  a  to 
/'"  gives  the  line  of  intersection  rebated  into  a 
horizontal  plane.  Then  proceeding  with  the 
construction  as  explained  at  Fig.  112  (it  will  be 


Fig.  113 


seen  that  Fig.  1 13  is  similarly  lettered),  the  angle 


}] 


iS6 


PRACTICAL  USES  OP 


I 


between  the  two  surfaces  of  the  lid  is  obtained. 
Half  this  angle  gives  the  bevel  required  for  the 
mitered  joint  as  there  shown. 
In  the  cases  considered  the  working  has  been 


Fig.  114 

in  front  of  the  vertical  plane  and  on  the  hori- 
zontal plane;  that  is,  in  the  first  dihedral  angle 
of  the  co-ordinate  planes.  This  has  been  the 
most  convenient  method,  because  the  surfaces 
slope  up  and  away  from  the  observer;  whereas 
in  the  cases  shown  by  Figs.  114  to  116  the  sur- 
faces slope  upward  and  toward  him,  and  there- 


i. 


THE  STEEL  SQUARE 


»57 


fore,  for  a  person  who  has  the  necessary  knowl- 
edge of  geometry,  the  simplest  method  will  be 
to  work  under  the  horizontal  plane,  but  in  front 
of  the  vertical,  that  is  in  the  fourth  dihedral 
angle  of  the  co-ordinate  planes.  Taking  the 
case  of  Fig.  114,  ab  is  the  plan  of  the  intersection 
of  the  two  surfaces.  At  right  angles  to  this,  set 
up  the  line  ^B,  making  it  equal  to  BC.  Join  aB; 
then  this  line  is  the  true  length  of  the  intersec- 
tion constructed  upward  into  the  horizontal 
plane.  Then  through  c  draw  de  at  right  angles 
to  ab — it  should  be  noticed  here  that  half  the  line 
de  coincides  with  B,  the  intersecting  surfaces 
being  equally  inclined.  Then  from  b  draw  bf  2X 
right  angles  to  ^ B.  Next,  with  f  as  a  center  and 
cj  as  radius,  describe  the  arc  cutting  ab  in  g. 
Then  joining  dg  and  eg  gives  the  bevel  required, 
as  shown  at  B.  By  imagining  the  object  at  Fig. 
1 14  turned  upside  down,  exactly  the  same  kind 
of  working  as  that  just  described  would  apply; 
but  perhaps  the  problem  will  be  simplified  by 
imagining  that  the  work  is  in  the  first  dihedral 
angle  of  the  co-ordinate  planes.  To  many  who 
understand  geometry  ihis  method  ot  working 
this  problem  will  commend  itself  as  being  sim- 
plest to  imagine,  although  giving  the  same 
results.     In  Fig.  115  precisely  the  same  reference 


I  'i 

i 

1    •      i| 


158 


PRACTICAL  USES  OF 


i 
■■I 


•    I 


.  -i 


letters  have  been  adopted,  and  it  will  be  seen 
that  the  same  problems  and  principles  are 
involved. 


Fui.  115 

The  geometrical  formula  for  finding  the  true 
shape  of  development  of  the  sides  is:  Given  the 
plane  and  inclination  of  a  plane  surface,  deter- 
mine its  true  shape.  A  prcblem  of  this  class  is 
shown  at  Fig.  117.  Let  adc(/e  be  the  plan  of  the 
given  figure,  its  side  ad  being  the  horizontal 
plane;  through  aS  draw  a  horizontal  line,  at 
right  angles  to  which  draw  .vy  as  shown;   next 


THE  STEEL  SQUARE 


'59 


set  out  the  angle  of  inclination  of  the  figure  as 
shown  hy  the  line  VT,  which  may  be  considerea 


Fio.  ii6 

as  the  vertical  line  of  a  singly  oblique  plane. 
From  the  plan  set  up  projectors  to  this  plane, 
then  rebate  back  the  figure  into  the  horizontal 
plane  by  drawing  the  arc  projections  to  xy,  and 
from  these,  projecting  at  right  angles  to  xy,  and 
from  cd  and  e  parallel  to  xy,  gives  the  points  CU 


^    4l 


t 


\\ 


.  ^  SB 


i6o 


PRACTICAL  USES  OF 


and  E.    Joining  a  to  E,  E  to  D,  and  D  to  C,and 
Kih  gives  the  true  shape  required. 


Fig.  tiy 

If  the  working  of  this  problem  has  been  care- 
fully followed  and  mastered,  its  direct  application 
to  cases  shown  at  Figs.  113,  114,  115,  and  ii6  will 
be  readily  understood. 

The  foregoing  is  meant  primarily  for  geome- 
tricians, and  secondarily  for  those  who  wish  to 
know  the  "reasons  why"  of  hopper  bevels. 

Having  shown  this  much,  simply  to  satisfy  the 
"learned"  in  "theoretical"  carpentry,  I  will  now 
proceed  to  show  how  all  the  lines  and  cuts  may 


THE  STKIiL  SQUARE 


i6i 


Fig  iiS. 


be  obtained  for  this  work  by  aid  of  the  square 
alone  and  in  a   speedier   and  simpler  manner. 

Let  us  suppose  Via. 
1 18  to  represent  a  sec- 
tion through  a  hopper; 
then  take  a  board  12 
inches  wide,  joint  one 
edge  and  draw  a  line 
of  the  side  elevation  according  to  the  refine- 
ments.    From  this  proceed  to  make  a  draft  with 

the  square  i  inch  to  a  foot. 
Take  12  inches  on  the 
blade,  hold  1 2  at  A,  and  by 
it  find  how  many  inches 
rise  the  given  inclination 
is  to  a  toot.  Draw  a  line 
by  the  tongue,  as  shown 
by  BV,  Fig.  119.  At  B 
draw  square  with  AC,  and 
again  square  on  line  AB 
through  C,  as  shown  by 
CL.  From  VB  erect  a 
perpendicular  also  to  C, 
cutting  VB  in  the  point  E. 
Fig.  119  By  these  several    opera- 

tions we  have  a  complete  draft  by  which  to  solve 
the  problem.    AB  is  the  given  slant,  and  has  9 


f 


I 


1  ! 


i 

i 


\  1 


i6a 


PRACTICAL  USES  OP 


inches  rise  to  a  foot.  AC  is  12  inches,  and  CB  is 
9  inches.  The  length  of  AB,  as  indicated  by  the 
figures  on  the  square,  is  15  inches.  Fifteen 
inches,  therefore,  is  the  width  of  the  board 
required  to  cut  the  hopper. 
Use  the  foot  draft  of  the  hopper  as  follows: 


Fig.  lao 
Take  on  the  blade  AB,  and  on  the  tongue  CB, 
and  apply  the  square  to  the  board  as  shown  in 
Fig.  120.     Mark  on  the  tongue,  which  will  give 
the  down-cut  bevel. 

For  a  butt  joint  take  EC  on  the  tongue  and 
CA  on  the  blade  and  apply  it  as  sh  -^wn  in  Fig. 
121.     Mark  on  the  square  edge  of  th«  Doard  EC. 

The  above  principles  will  give  the  manner  of 
backing  a  hip  rafter.  Suppose  that  AC,  Fig. 
119,  were  the  seat  of  the  hip  rafter,  and  AB  the 


li  I*. 


THE  STEEL  SQUARE 


163 


length  of  the  hip,  by  taking  CL  on  the  tongue 
and  CB  on  the  blade,  marking  on  CL  and  setting 


Fig.  121 

the  bevel  by  that  line,  the  exact  backing  of  the 
hip  will  be  obtained.  The  handle  of  the  bevel  is 
to  be  square  across  the  hip  rafter,  as  shown  at 
CL  in  Fig.  119. 

A  very  good  way,  and  one  I  have  found  to 
work  out  correct,  and 
which  I  take  from 
"Carpentry  and 
Puilding,"  for  obtain- 
ing the  butt  joints  for 
hoppers,  is  as  follows: 
Draw  a  line  CD,  Fig. 
122,  at  the  same  an- 
gle with  the  straig  h 
edge  of  the  board  ^ 
AB  as    the  sides  of  fig.  122 


i   . 
i 


•  *i 


i<4 


PRACTICAL  USES  OP 


the  hopper  are  to  stand.  Cut  this  line  at  any 
point,  E,  with  a  line,  FG,  drawn  at  right  angles 
with  AB.  Divide  CF  into  five  equal  parts,  and 
through  the  fourth  point  thus  established  draw  a 
line  as  shown  from  H,  cutting  E.  Then  HE  will 
be  the  bevel  of  the  butt  joint-  This  rule  will  work 
no  matter  what  the  flare  of  the  side  or  width 
of  boards  used,  providing  always  the  hopper  is 
square-cornered. 


Fi  ■  I .  I  >  I  ■  I  ■  I .  I  ■  I ,  I ,  I  ■  i>SM  ■  I ,  I  ■  I .  I .  I  ■  I  ■  I  ■  I .  I ,  I , 


Fig.  123 

A  handy  rule  is  given  as  follows:  Let  Fig.  123 
represent  the  slant  of  the  hopper  as  shown  by 
the  line  running  from  12  to  12  on  the  square. 
Figs.  124,  125,  and  126  show  the  applications. 
Fig.  123  represents  the  square  with  a  line  drawn 
from  12  to  12;   this  line  shows  the  flare  of  the 


Tte. 


THE  STEEL  SQUARE 


165 


hopper.    The  distance  from  12  to  12  on  the  diag- 
onal line  is  17  inches  nearly,  as  I  have  shown 


Fig.  124 


previously.    The  angle  of  the  miter  on  the  hori 


Fig.  125 


zontal  is  45°  or  a  true  miter.     The  base  line 


'   ^.i•^ 


;    I: 


"n 


■^^ 


^Mtfttl 


li 


ll 


i66 


PRACTICAL  USES  OF 


ii 

i 
i 


from  the  corner  of  the  square  to  12  is  t  2  inches, 
the  pitch  line  is  5  inches  longer  tha-  the  ^)ase 
line  (the  difference  b**»^wcen  17  and  i  2  inches). 
Add  5  inches,  the  excess  of  the  pitch  from  the 
base  line,  and  we  have  12  and  17  inches  of  bevel 
and  miter.  The  application  of  this  is  shown  in 
Figs.  124  and  125.  For  a  butt-joint  the  sides  slant 
two  ways  on  each  angle.  The  angle  on  the  hori- 
zontal is  a  right  angle.  The  excess  of  the  pitch 
from  the  base  line  on  the  one  side  is  5  inches, 
and  on  two  sides  10  inches,  making  an  angle  or 


Fig.  T26 

joint  12  and  10,  as  shown  in  Fig.  126.  The  lat- 
ter figure  shows  the  application  of  the  rule 
named;  all  the  stuff  being  square-edged.  This 
method  when  thoroughly  understood  is  very  sim- 
ple and  effective. 
Another  method,  somewhat  similar  to  the  one 


THE   STEEL   SgUARE 


.67 


already  shown,  is  given  herewih.  To  obtain 
the  bevels  of  a  hopper  either  by  the  square  or 
lines,  the  rise  to  the  foot  of  the  sides  being  given, 
first  ascertain  th'  hypothenuse  from  12,  taken 
on  the  blade  -f  th'-  sqi:ar(  ,  to  the  rise  to  the 
foot,  taken  on  the  tongue.  1  )ivi(le  the  square  of 
12  by  the  rise  to  the  loot.  Apply  the  squan  to 
a  straight -edge,  taking'  the  hypothenuse  on  the 
blade  and  ^  on  the  tonj;u( .  This  will  denote 
the  surfuir  lx.v.  .  .">' tin.  apply  the  square  to 
the  straight-edge  taking  the  hypothenuse  on  the 
blade  and  the  rise  to  the  foot  on  the  tongue. 
This  will  denote  the  bevel  of  the  miter-joint,  so 
called.  For  a  butt-joint,  rake  on  the  blade  the 
quotient  arising  from  the  division  of  the  square 
of  12  by  the  rise  to  the  foot,  and  on  the  tong 
take  the  hypouienuse.  Then  the  blade  wi.i 
denote  the  bevel  required. 

Another  methxl  on  the  same  diagram  is  .as 
follows:  Apply  the  square  to  a  straight-edge, 
taking  12  on  the  blade,  the  rise  to  the  foot  on 
the  tongue,  and  mark  by  the  blade  to  obtain  AD 
of  Fig.  127,  which  represents  the  inclination  of 
the  sides.  At  random  make  BD  perpendicular 
to  .ABC.  which  represents  the  straight-edge, 
taking  AD  on  the  blade  and  AB  on  the  tongue. 
This  will  give  the  surface  bevel.    Again,  apply 


I ,  m 


i 

i 

,     ii 
i| 

'      il 


II 


;  k 


f 


; 


llii 


ii 


Its 


PRACTICAL  USES  OF 


the  square,  taking  CD  on  the  blade  and  BC  on 
the  tongue,  which  will  denote  the  bevel  for  the 


miter-joints.  For  butt-joints  apply  AB  taken  on 
the  blade  and  CI)  taken  on  the  tongue.  The 
blade  will  denote  the  bevel  required.  The  same 
results  may  be  obtained  by  geornetrical  construc- 
tion, as  follows:  Referring  to  Fig.  127,  make 
DF  equal  to  AB,  and  draw  AF;  make  DG  equal 
to  CD  and  draw  FG;  make  CE  perpendicular 
to  CD  and  equal  to  BC  and  draw  DH.  The 
designated  acute  angles  at  F  will  be  the  angle  of 
the  surface  bevel.  The  designated  obtuse  angle 
at  F  will  be  the  bevel  for  the  butt-joint.  The 
angle  at  F  is  the  bevel  for  the  miter-joint. 

The  foregoing  may  be  applied  to  roofs  of  one 
pitch  over  rectangular  bases.  Fig.  128  repre- 
sents a  section  of  cornice.  That  which  relates 
to  the  surface  bevel  is  applicable  to  the  surface 
bevel  of  the  boarding,  the  outward  bevel  of  pur- 


i    i      i  i 


THE  STEEL   SQUARE 


169 


Pig.  138 


lins  which  come  in  contact  with  each  other  or 
with  hip  rafters,  the  surface  bevel  of  the  planceer 
in  a  cornice  similar  to  _, 

the  diagram,  and  the  V^^^^^^^^^^^ 
edge  bevel  of  a  fascia.  f^~^ 
That  concerning  miter- 
joints  is  applicable  to 
the  edge  bevel  of  the 
boarding,  the  inward 
or  down  bevel  of  purlins,  and  the  surface  bevel 
of  the  fascia.  Applying  the  square  to  a  straight- 
edge, according  to  the  directions  given  in  the 
first  and  second  methods  above  presented  for 
obtaining  the  surface  bevel,  and  marking  by 
the  blade,  we  obtain  the  edge  bevel  of  jack 
rafters.  In  Fig.  127,  FAD  denotes  the  edge 
bevel  of  jack  rafters.  In  order  to  properly  get 
the  crown  molding  for  a  cornice  similar  to  the 
drawing,  lay  off  on  the  edge  of  a  miter-box  the 
surface  bevel  of  the  board,  and  on  the  sides  the 
edge  bevel. 

Hopper  cuts  to  a  large  extent  are  similar  to 
the  cuts  required  for  fitting  boards  in  or  over 
;i  valley  or  hip  roof;  consequently  the  figures 
on  the  square  that  give  the  cuts  for  the  roof 
boards  must  give  the  cut  for  a  hopper  of  ti:e 
same  pitch. 


si 
/] 


Y  1 

1       1] 


1 .    ;.( 


17© 


PRACTICAL  USES  OF 


■       I; 


I        -.i 


i 


Fig.  129  shows  a  hopper  in  different  views,  as 
follows:    Beginning  at  the  top  is  the  top  view  of 

171 


Fin.  129 


I 


the  hopper.  As  far  as  this  part  is  concerned  all 
hoppers  look  alike,  as  there  is  nothing  in  this  to 
distinguish  the  pitch.     .\ext  is  the  sectional  or 


THE  STEEL   SQUARE 


»7i 


side  view.  In  this  is  shown  the  thickness  of  the 
boards  and  the  flare  or  pitch,  which  in  this  is  the 
three-quarter  pitch.  Following  (Fig.  129)  is 
shown  the  four  sides  in  the  collapsible  or  ready 
to  be  p  It  together,  followed  with  the  top  view  of 
the  edge  of  the  board. 

Of  course  it  is  not  necessary  to  lay  out  all  of 
this  diagram,  or  any  of  it  for  that  matter;  it  is 
done  by  way  of  illustration.  See  the  application 
of  the  square,  which,  in  this  case,  is  12  and  215^. 
But  why  use  these  numbers?  Because  the  flare 
given  is  the  three-quarter  pitch,  or  12  and  18  on 
the  square,  and  the  hypothenuse  of  these  num- 
bers is  2154,  the  tongue  giving  the  side  bevel. 
When  working  full  scale  it  is  always  12  on  the 
tongue. 

For  the  miter  bevel,  the  top  edges  should  be 
first  beveled  so  as  to  be  level  when  in  position. 
The  miter  would  be  at  an  angle  of  45°,  and 
any  of  the  equal  numbers  on  the  square  gives 
this  cut;  but  if  the  edges  are  to  be  left  square 
with  the  sides,  as  shown,  the  above  will  not 
work. 

To  accomplish  this,  however,  a  very  simple 
way  is  shown  in  diagram  at  A,  or  in  Fig.  130  as 
follows:  Lay  off  ^he  base  and  the  desired  pitch, 
and  on  the  latter  measure  the  thickness  of  the 


Gj 


^H 


■"SSL-W^Effw 


i 

I!" 


17a 


PRACTICAL  USES  OF 


board  as  at  AB.     From  B  draw  a  plumb  line  to 
base.     BC  is  the  width  apart;    the  side  bevels 

should  be  along  the 
edge  of  the  board. 

In  case  of  large 
hoppers  to  be  built 
to  suit  some  particu- 
lar place,  or  regard- 
less of  pitch,  it  is  bet- 
ter  to  use   the  one- 


FiG.  130 


inch  scale  as  illustrated  in  Fig.  131.    The  figures 
"jYi  and  ijj^  give  the 
side    cut,   while    the 
section  gives  the  mi- 
ter. 

This  diagram  is  all 
that  is  necessary  to 
find  the  cuts  for  any 
size  hopper,  and  were 
it  not  fi)r  the  miter 
even  this  is  unneces- 
sary. 

Another  method  is 
here  offered  which  is 
taken    from   the   old  F"'*"-  mi 

"American    Builder,"    and   which    many    of   the 
older  workmen  will  recognize,  as  it  was  at  one 


■  n 

-u 

-ti 

-M 

-1* 

'74 

-17 

-V 

-It 

■  M 

-II 

-« 

f 

"?■ 

-s 

i^/ 

-» 

€"•' 

-T 

;:< 

-t 

* 

IK 

-S 

1 

' 

-1-; 

.... 

K 

V'. 

■  1 

>     A    «    «    * 

1    T    1    t    1 

1 : 1   1   1   1 

:r\ 

hoNVV 

7i 

THE  STEEL  SQUARE  173 

time  in  great  favor.     The  lines  E,  D,  H,  A,  ond 


i  m 


h: 


>74 


PRACTICAL  USES  UF 


C,  G,  B,  Fig.   132,  show  the    outside   edges   of 
the  steel   square,  or  squares;    OO  is    the  edge 
of  the  board.      Half  the  width  of   the   width 
of  the  top  less  half  the  discharge  hole  on  the 
blade  AB.    The  depth  on  the  tongue  BC  gives 
the  diagonal  AC,  the  width  of  the  side.     Makt- 
AD  on  the  blade  equal  AC,  and    DE   on  the 
tongue  equal  AB,  and  E  is  the  bevel  for  face  of 
stuff.     Make  GB  equal  FB;   connect  GA,  and  G 
is  the  miter  cut  for  edge  of  stuff.     Square  a  line 
from  00,  cutting  AB.     To  save  extra  lines  use 
FB.    Make  AH  (on  AD)  equal  AI-.     HK  square 
from  AD,  and  FM  equal  HK;   connect  MB,  and 
M  is  the  bevel  for  straight  cut  (to  nail  on,  as  we 
nail  a  box  together,  square  instead  of  miter),  the 
long  point  inside.     If  we  make  the  line  cutting  at 
H  square  from  AE  instead  of  AD,  as  XH   and 
FS  equal  NH,  connect  SB,  and  S  is  the  trying 
bevel  for  straight  cut.     Bisect  BSO;   draw  the 
miter  line  and  it  gives  P  the  trying  bevel  for 
miter  cut;  the  lines  are  extended  and  bevel  P 
placed  below  the  line  for  want  of  room  above 
And  yet  another  method:     Suppose  ABCD, 
Fig-  i33>  to  represent  the  elevation  of  a  box,  the 
sides  of  which  have  a  slope  of  45°.    Take  C  as  a 
center  and  CB  as  radius,  and  describe  the  semi- 
circle  under  the   line   AB,  as  shown   by   Hi:j. 


THE  STEEL  SQUARE  ,„ 

From  C  let  fall  a  perpendicular  through  the  line 
AB,  intersecting  the  semicircle  in  the  point  E. 
Through  the  point  E 
thus  established  draw 
a  horizontal  line  or  a 
line  parallel  to  AB,  *; 
making  it  in  length 
equal  to  AB.     From  F'«  133 

the  point  F  thus  established  draw  FC.  Then  the 
bevel  at  F  will  be  the  bevel  to  be  used  for  the 
sides  of  the  box  or  hopper.  To  find  the  bevel  of 
miters  at  the  corners  with  K  as  center  and  radius 
equal  to  the  distance  from  K  to  the  line  CB,  de- 
scribe an  arc  cutting  the  line  KC,  thus  establish- 
ing the  point  L.     From  L  draw  the  line  LJ,  cut- 


Fio.  134 

ting  the  line  AB  at  the  point  J,  which  represents 
the  intersection  of  the  arc  first  drawn  with  the 
AB.    Then  the  bevel  at  L  is  the  bevel  of  the  line 


I 


if 


I 


t  :l 


i>! 


176 


PRACTICAL  USES  OF 


miters  at  the  corners.     Fig.  134  shows  the  sides 
of  the  box  laid  out  flat. 

Mr.  Woods  recently  published  in  the  "The 
National  Builder,"  a  paper  on  bevels  and  hop- 
per cuts  generally,  which,  owing  to  its  excel- 
lency, I  have  thought  worthy  of  a  place  in  this 
work.  The  author  starts  out  by  saying:  "If 
there  was  no  given  pitch,  then  the  sides  would 
be  vertical.  Now  with  12  on  the  tongue  as  cen- 
ter, draw  an  arc  of  same  radius  from  the  heel  to 
a  point  directly  above  and  square  over  the  blade; 


Fig.  135  I'^if-.  1-56 

12  and  i2willgi\«  the  miter.  Fig.  135.  Simple 
enough,  but  do  you  know  that  this  simple  ruk- 
applies  when  there  is  a  pitch  given?  At  thr 
point  where  the  am  intersects  the  pitrh  taken  on 
the  blade  will  give  the  miter,  the  blade  giving 
the  c"t     See  Fig.  136. 


77 


THE  STEEL  SQUARE 

"For  the  side  bevel  across  the  board,  transfer 
the  length  of  the  pitch  to  the  blade,  the  tongue 
giving  the  cut.  These  figures  also  give  the  side 
cut  of  the  jacks  for  a  roof  of  same  pitch.  The 
blade  in  this  case  giving  the  cut. 

"Now  wc  will  give  another  method  of  finding 
the  miter.  In  all  roofs  and  hoppers  there  is  an 
unseen  pitch  which  we  will  call  co-pitch.  Assum- 
ing that  the  edges  of  our  boards  are  square  the 


^4ia* 


Fig.  137 


co-pitch  v,-ould  stand  at  an  angle  of  qo'  with  the 
given  pitch."     See   Fig.  137 

The  rule  given  in  Fig.  136  for  the  side  oevel 
will  apply  to  the  miter,  but  instead  of  using  the 
length  of  the  given  pitch  substitute  that  of  the 
co-pitch,  and  by  referring  to  Fig.  137  we  find  this 


■ 


It 


' 


I 


nil 


•i 
ii 


n 


i 


178 


PRACTICAL  USES  OP 


to  be  IS'I  inches  on  the  blade.     No\ "  for  proof, 
see  Fig.  138.    Twclvt?  and  gyi  first  method,  and  12 


Fig.  138 

and  15K  second  method.    The  blade  giving  the 
cut  in  the  former,  and  the  tongue  in  th(  latter. 


15;^  Miter 

'.'6 

2l|  Butt  Miter 


r'O.    ly, 


THE  STEEL  SQUARE 


«79 


In  Fig.  139  is  shown  all  that  is  contained  in  the 
other  figures  and  then  some  more.  In  this  we 
run  up  against  another  pitch.  It  is  an  extenua- 
tion of  the  co-pitch  to  a  point  on  a  level  with  the 
starting  point  of  the  given  pitch.  This  we  will 
call  complement  pitch.  The  length  of  this  pitch 
transferred  to  the  blade  will  give  the  butt-joint 
or  miter;  the  tongue  giving  the  cut. 

In  Fig.  140  is  shown  a  dia- 
gram for  a  hopper  of  one- 
half  pitch.  In  this  all  of  the 
pitches  are  of  equal  length.  < 
Therefore  12  and  163 1  will 
give  all  of  the  cuts;  the 
tongue  giving  the  cuts  in 
each  case.  Figs.  139  and 
142  are  simply  fillers  and 
self-explanatory. 

While  editor  of  the  "  Builder  and  Woodworker" 
of  New  York,  I  had  considerable  communication 
with  the  late  Robert  Riddell  whose  works  in  the 
6o's,  7o*s,  and  8o's  were  very  popular,  and  on  one 
occasion,  in  1879.  he  sent  to  me  the  following 
diagrams  and  explanation  of  a  method  for 
obtaining  the  bevels  and  cuts  for  nearly  all  kinds 
of  flared  work  or  inclined  framing.  This  prob- 
lem and  solution,    said  Mr.  Riddell,  is    offered 


Fig.  140 


-'   ''H 


MICROCOPY   RESOLUTION   TEST   CHART 

(ANSI  and  ISO  TEST  CHART  No.  2) 


1.0 


I.I 


1.25 


1^  ill  2.8 

IIIIM 

1^  !■■     112  2 

If   ti^       '^ 

1  -^  lll^-° 

»-     ^ 

KbL^l- 

1  ''^ 

1.4 

1.6 

^  /APPLIED  IIVMGE     Inc 

=;  1653  East    Mam   Street 

r.=  "Rochester.    Ne»   Vork        14609       USA 

JS  (716)   482  -  0300  -  Phone 

1^  (716)  288  -  5989  -  Fo« 


^ 


I  l' 


^ 


1 


11  '  '  ::  i 

111 


i8o 


PRACTICAL   USES  OF 


for  the  first  time  for  publication,  as  a  reliable 
and  simple  method  for  obtaining  bevels  and  cuts 


Fig.  141 


for  nearly  all  kinds  of  inclined  framing,  and  for 
the  finding  of  cuts  for  splay  and  flare  work. 


FiQ.  143 


The  diagrams  show  how  the  cuts  for  hopper 
work  may  be  obtained   for   angles   other  than 


m 


THE  STEEL  SQUARE 


i8i 


right  ones,  also  for  getting  angles  for  corner 
posts  having  a  double  inclination. 


Fig.  143 

Let  us  suppose  the  plans  shown  at  Figs.  143  and 
144  to  be  right  angle  figures,  having  sides  which 
incline  or  flare  equally  to  any  desired  angle.  A 
corner  post  is  also  used  which  will  incline  same 
as  the  ends  and  sides.  The  junction  at  the 
angles  may  be  either  mitered  or  a  butt,  as  either 


iU 


i8a 


PRACTICAL  USES  OF 


i  ; 


Style  of  joint  may  be  obtained  when  desired.    To 
describe    the    problem,  begin  by  drawing  two 


R 
Fig.  144 

oarallel  lines,  AB  an  '  DC,  any  reasonable  dis- 
tance apart.  Assume  AC  as  inclination  or  flare 
of  sides.    From  N  square  down  a  line  making 


mBm\i\s^Ki-xi.  '^^'^am'Z'-'-wr^ 


>j<, 


VjFr* 


THE  STEEL  SQUARE 


183 


NA  and  NR  equal.  From  A  square  down  a  line 
cutting  D;  join  RD,  and  in  the  angles  thus 
formed  find  bevel  2  for  cut  on  face  of  sides 

To  find  the  bevel  for  miter  on  edge  of  stuff, 
take  N  as  a  center  and  describe  an  arc,  touching 
the  line  AB  and  terminating  at  J.  From  B  draw 
a  line  through  J  indefinitely.  This  gives  bevel  3 
for  the  miter. 

To  find  the  corner  post,  proceed  as  follows: 
Make  NC  equal  XD;  join  RC,  and  extend  AN 
to  cut  RC  at  P,  from  which  square  up  a  line  cut- 
ting at  B.  From  N  draw  through  B,  thus  form- 
ing both  angles  of  the  corner  post,  and  giving 
bevel  4,  which  answers  either  for  a  butt-joint  or 
the  shoulder  cuts  on  cross-rails  of  framing. 

Nothing  can  be  more  simple  or  more  accurate 
than  this  method,  and  it  may  be  easily  tested  by 
first  drawing  the  "spread  out,"  as  shown  on  the 
upper  part  of  the  diagram  on  cardboard,  and 
cutting  through  in  the  lines  marked  X,  X,  X, 
then  fold  on  the  lines  marked  O,  O,  O. 

Bring  the  points  S,  S  together,  and  the  mode 
of  construction  will  readily  be  understood. 

The  flare  may  be  any  angle,  the  results  will 
be  the  same. 

Another  problem  and  solution,  and  I  am  done. 
The  problem  I  am  now  about  to  present  is  one 


l-.jr 


wsiaiics^A-^w.'m^^: 


i84 


PRACTICAL  USES  OF 


that,  when  thoroughly  grasped  by  the  workman, 
will  enable  him  to  lay  out  the  lines  for  every 
cone  'vable  cut  required  in  tapered  framing, 
paneling,  or  splayed  work  of  any  kind  when 
angles  are  thrown  out  of  square.  To  Peter 
Nicholson  is  due  the  credit  of  this  method  of 
solution,  but,  as  in  many  other  things,  ihe  master 
hand  of  the  late  Robert  Riddell  improved  and 
simplified  it  and  clothed  it  in  language  easily 
understood  by  the  American  workman. 

Let  us  suppose  the  line  AD,  at  Fig.  145,  to  be 


\ 


Fig.  145 

a  given  base  line  on  which  a  slanting  side  of 
hopper  or  box  rises  at  any  angle  to  the  base 
line,  as  CB,  and  the  total  height  of  the  work  is 
represented  by  the  line  BE.  By  this  diagrara  it 
will  be  seen  that  the  horizontal  lines  or  bevels  of 
the  slanting  sides  are  inaicated  by  the  bevel  Z. 
Having  got  this  diagram,  which,  of  course,  is 
not  drawn  to  scale,  well  in  hand,  the  ground  plan 


IJ 


THE  STEEL  SQUARE  ..^s 

of  the  hopper  may  be  laid  down  in  such  a  shape 
as  desired,  with  the  sides,  of  course,  having  the 
slant  as  given  in  Fig.  145. 

Take  T-*,  3S,  Fig.  146,  as  a  part  of  the  plan, 
then  set  d!  the  width  of  sides  equal  to  CB,  as 
shown  in  Fig.  145.  These  are  shown  to  intersect 
at  PL  above;  then  draw  lines  from  PL  through 
2-3  until  they  intersect  at  C,  as  the  dotted  lines 
show.  Take  C  as  a  cenrer,  and  with  the  radius 
A  describe  the  semicircle  AA,  and  with  the  same 
radius  transferred  to  (  Fig.  145,  describe  the 
arc  AB,  as  shown.  Again  with  the  same  radius, 
set  off  arcs  AB,  AB  on  Fig.  146,  cutting  the  semi- 
circle at  B,  as  shown.  Now  draw  through  B,  on 
the  right,  parallel  with  S3,  cutting  at  J  and  F; 
square  over  FH  and  JK,  anu  join  HC;  this  gives 
bevel  X  as  the  cut  for  face  of  sides  which  come 
together  at  the  angles  shown  at  3.  The  miters 
on  the  edge  iff  are  parallel  with  the  dotted 

line  L3,  This  is  the  acute  corner  of  the  hopper, 
and  as  the  edges  are  worked  off  to  the  bevel  Z, 
as  shown  in  Fig.  145,  the  miter  must  be  correct. 

Having  mastered  the  details  of  the  acute 
corner,  the  square  corner  at  S  will  be  next  in 
order.  The  first  step  is  to  join  KV,  which  gives 
the  bevel  Y  for  the  cut  on  the  fac(  of  sides  on 
the  ends  which  form  the  square  corners.     The 


m 


VM 


^p 


I 


m  ;  ii  i  * 


m 


i86 


PRACTICAL  USES  OF 


Pig.  146 


Si- 


rmwv'mmma^rs^Md^^sr3SN?Mis^mx^*'^r?!F.,wmsmAm.  * 


:(4'*itvJ!»-.^ 


II » 


THE  STEEL  SQUARE 


187 


method  of  obtaining  these  lines  is  the  same  as 
that  explained  for  obtaining  them  for  the  acute 
angled  corner  as  shown  by  the  dotted  lines.  As 
the  angles  S  T  are  both  square,  being  right  and 
left,  the  same  operation  answers  both;  that  is, 
the  bevel  Y  does  for  both  corners. 

Coming  to  the  obtuse  angle  P2,  we  draw  a  line 
BE  on  the  left  parallel  with  A2,  cutting  at  E,  as 
shown  by  dotted  line.  Square  over  at  E,  cutting 
TA2  at  N;  join  NC,  which  will  give  the  bevel 
VV,  which  is  the  an^le  of  cut  for  face  of  sides. 
The  miters  on  edges  are  found  by  drawing  a  line 
parallel  with  P2. 

In  this  problem,  like  the  orevious  one,  every 
line  necessary  to  the  cuttin.  of  a  hopper,  after 
the  plan  as'  shown  by  the  boundary  lines  2,  3,  T, 
S,  is  complete  and  exhaustive;  but  it  must  be 
understood  that  in  actual  work  the  spreading  out 
of  the  sides,  as  here  exhibited,  will  not  be  neces- 
sary, as  the  angles  will  find  themselves  when  the 
work  is  put  together.  When  the  plan  of  the 
base— which  is  the  small  end  of  the  hopper  in 
this  case — is  given,  and  the  slant  or  iuclination 
of  the  sides  known,  the  rest  may  easily  be 
obtained.  In  order  to  become  thoroughly  con- 
versant with  the  problem,  I  would  advise  the 
reader— as  has  often   been   before  advised  by 


HI 


If 


'''^tm^  txvt' 


Sd  ■r"i«<fi.:fcBi 


■^ 


1 88 


PRACTICAL  USES  OF 


teachers — that  it  would  be  well  to  have  the  draw- 
ing made  on  cardboard,  so  as,  by  cutting  out  all 
the  outer  lines,  including  the  open  corners  which 
form  the  miters,  to  leave  the  whole  piece  loose. 
Then  make  slight  cuts  in  the  back  of  the  card- 
board, oppo<^ite  the  lines  2,  3,  S,  T,  just  deep 
enough  to  admit  of  the  cardboard  being  bent 
upward    on    the    cut    lines    without    breaking. 
Then  run  the  knife  along  the  lines  which  indi- 
cate the  edges  of  the  hopper  sides.     This  cut 
must  be  made  on  the  face  of  the  drawing,  so  as 
to  admit  of  the  edges  being  turned  downward. 
After  all  the  cuts  are  made,  raise  the  side  until 
the  corners  come  closely  together  and  let  the 
edges  fall  level,  or  in  such  a  position  that  the 
miters  come  closely  together.     If  the  lines  h'  -e 
been  drawn  accurately  anu   he  cuts  made  on  .   e 
lines  in  a  proper  manner,  the  work  will  adjust 
itself  nicely  and  the  sides  will   have  the  exact 
inclination   shown   at    F"ig.    145,  and   a   perfect 
model  of  the  work  will  be  the  result. 

Before  leaving  the  subject  of  "Bevels,  Splays, 
and  Hoppers,"  I  wish  to  call  the  attention  of  my 
readers  to  Figs.  143  and  144,  in  order  that  they 
may  get  a  clear  understanding  of  the  manner  of 
getting  the  bevels  for  corner  posts  for  hoppers. 
The  methods  of  determining  the  proper  bevels 


»?w!;rw^*yife,;^>"i''3£!-?:- 


L-Tff.-xdfsi^Ei^s^!  TTifl  w?«iP«3?;-T'':i*»iaap^ri 


THE  STEEL  SQUARE 


189 


ana  anpfles  for  those  posts  me  set  forth  at  con- 
siderable length  in  the  text  and  drawings,  and 
what  I  desire  to  show  is  that  the  corner  pcit  of  a 
hopper  is  exactly,  in  miniature,  similar  to  the 
corner  post  of  a  frame  building  having  a  double 
inclination;  or,  in  other  words,  a  tapered  build- 
ing, with  the  exception  that  a  hopper  has  its 
smallest  area  as  a  base,  while  a  tapered  building, 
a  pyramid,  has  its  largest  area  as  a  base. 

Now  it  must  be  evident  that  the  lines  giving 
the  proper  angles  and  bevels  for  the  corner  post 
of  a  hopper  must  of  necessity  give  the  proper 
lines  for  the  corner  post  for  a  pyramidal  building, 
such  as  a  railway  tank  frame,  or  any  si  /.ilar 
structure.  True,  the  position  of  the  post  is 
inverted,  as  in  the  hopper  its  top  falls  outward, 
while  in  the  timber  structure  the  top  inc'  u  ^ 
inward;  but  this  makes  no  difference  it  the 
theory,  all  the  operator  has  to  bear  in  mind  is 
that  the  hopper  in  this  case  is  reversed.  Once 
the  proper  shape  of  the  corner  post  has  be^n 
obtained,  all  other  bevels  can  readily  be  found, 
as  the  side  cuts  for  joists  and  braces  can  be  taken 
from  them.  A  study  of  these  two  figures  in  this 
direction  will  lead  the  student  up  to  a  correct 
knowledge  of  tapered  framing. 

I  leave  the  subject  of  splays  and  bevels  here. 


!  ij 


M 


^'ZPfsmvA  ■  -\^  -  wjFwmr^^^ea^^^arr-mmm^^t^simKFivEmiMaeEBiaBcmj^is^iFr 


i   !; 


■mi 


i 
1  ;; 


'li 


i| 

II 

mm  ^ 

-'  fef 

190 


PRACTICAL  USES  OP 


but  I  may  add  that  I  have  not,  by  any  means, 
shown  all  the  ways  and  means  of  fmdinjr  solu- 
tions of  problems  presented,  but  I  am  persuaded 
the  examples  I  have  set  forth  are  the  best  suited 
for  the  workmen  of  this  country  because  of  tlicir 
simplicity  and  of  the  manner  in  which  they  have 
been  laid  down.  I  think  every  possible  kind  of 
hopper  and  splay  has  been  touched  upon,  and,  if 
not,  I  am  sure  the  rules  given  will  enable  any 
workman  who  has  followed  me  closely  to  deal 
with  the  difificulty  successfully. 

At  Fig.  147,  I  show  one  of  the  usual  methods 
of  finding  the  circumference  of  aciicle;  this  is 
c 


Fig.  147 

done  by  taking  the  radius  AB  on  the  compass, 
and  with  E  as  a  center  intersect  the  curve  in  U; 
then  with  the  same  radius,  and  C  as  a  center. 
draw  the  arc  from  A,  making  it  and  DC  equ  .1; 


mi'miiS'G^m'fm 


|?TJ 


THE  STEEL  SQUARE 


to. 


draw  from  C,  througl  the  point,  cutting?  in  2. 
which  jjives  2K  as  one-fourth  of  the  circumfer- 
ence of  a  :ircle  havinjr  AB  as  a  radius.  The 
same  rest  can  be  obtained  almost  instantly  by 
using  a  .  i-square  having  an  ,'le  of  60  and 
with  it  draw  a  line  from  C  I  "  ii  will  cut  the 
diameter  at  2. 

It  is  generally  supposed  that  this  method  is  a 
correct  one,  or  at  least  sufficiently  so  for  prac- 
tical purposes.  It  is  not  so,  however,  and  when 
used  sometimes  leads  to  error  and  trouble. 

A  much  more  accurate  method  is  shown  at 
Fig.  148,  which  is  obtained  as  follows:  Take  A 
as  center  and  with  B  as  a  radius,  describe  the 
arc,  and  from  A  with  an  angle  of  45°,  draw  a  lin  . 
cutting  at  E;  w  the  same  angle  draw  BJ;  from 
J  draw  sq  lare  \^.in  AB  cutting  the  arc  in  2;  join 
it  with  E.  .hen  four  times  2E  wil!  be  found  to  be 
i  '  -a!  to  tie  circumference  of  a  circle  having  the 
laaius  AB. 

Suppose  the  diameter  2B  of  the  circle  shown 
at  Fig.  149,  to  be  16  feet  in  diameter;  make  a 
similar  figure  or.  a  scale  of  one-quarter  of  an 
inch  to  the  foot,  then  the  exact  circumference  of 
this  may  be  found  by  drawing  AE  with  the  angle 
45°;  then  with  the  same  angle  draw  BR,  and  fn  a 
R  draw    quare  with  AB  cutting  the  circle  in  D; 


:i  m 


il' 


-  i  ] 

I  i 

►  I] 


i>pan9v<v?"vs'iK,  vauiiBES^^msmm^Tm 


MW^^^*^«EWMPre^^55»^ 


193 


PRACTICAL  USES  OP 


join  it  and  E,  and  four  times  this  line  DE  will  be 
equal  to  the  circumference  of  the  circle.  This 
may  be  proven  as  follows:  Join  B,  E,  then  draw 
E  parallel  with  the  diameter  cutting  the  perpen- 
dicular in  J;  draw  through  R  parallel  with  JE 
cutting  in  F  and  H;  join  J,  D,  and  we  have  AB, 
which  divided  the  circumference  into  six  equal 
parts;  i.e.,  DC7,  BE8,  FH9,  ACio,  KJii,  NP12, 
BL13,  PR14,  and  BC  into  16  parts.  This  dia- 
gram also  gives  the  lengths  of  chords  which 
divide  the  circle  into  equal  parts  and  proves  the 
accuracy  of  the  line  DE,  a  four  times  this  line 
will  be  the  circumference. 

Now  comes  in  the  steel  square,  where  these 
dimensions  may  be  worked  out  without  drawing 
a  line.  Looking  at  Fig.  149  we  find  that  the 
lines  DJ  and  UR  form  an  angle  in  which  is  a 
bevel  marked  X  vhich  plays  an  important  part 
in  the  next  illustration. 

Let  EB  on  the  blade  of  the  square,  shown  at 
Fig.  150,  be  the  radius  of  the  circle;  lay  a 
straight-edge  across  the  square,  keeping  one 
edge  on  the  point  E,  then  apply  the  bevel  X,  as 
shown,  to  the  blade  of  the  square  so  that  the 
blade  of  the  bevel  will  align  with  the  edge  of  the 
straight-edge;  then  the  distance  BR  on  the 
blade  will  be  twice  the  length  of  DE,  Fig.  149, 


l'_/-Vr-r-     ,lir  r  .■'.■:Jm^. 


k--lm 


*':  i 


THE  STEEL  SQUARE 


m 


and  consequently  equal  to  half   the  circumfer- 
ence of  the  circle. 


Fig.  149 

To  find  the  figures  on  the  square  that  shall 
give  an  angle  equal  to  that  of  DJE,  Fig.  149,  in 
which  is  seen  that  bevel  X,  lay  a  straight-edge 
across  the  square  keeping  the  edge  to. the  mark 
gj4  on  the  blade  and  3  on  the  tongue;  then  the 
angles  thus  formed  by  the  straight-edge  and  the 
square  will  be  found  to  exactly  agree  with  that 
of  bevel  X,  which  was  obtained  by  the  geomet- 


ii- 
11:- 


•1 ; 


^11 


«?PT 


mm 


194 


PRACTICAL  USES  OP 


Fi3.  150 


'Ti.  :-:-<ir.«u 


■t»-"jr  :.  ',^'1  "?i*?L-.T ■■ 


•'.  .-..-.'flJL.P 


;ifl 


'HE  STEEL  .SQUARE 


«95 


Heal  construction  of  Fig.  149.    This  is  a  useful 
problem  and  may  often  be  put  to  practical  use. 

To  divide  a  line  of  any  length  into  any  number 
of  equal  parts  by  the  steel  square,  we  proceed  as 
follows:  The  heel,  or  external  angle  of  the 
square,  is  shown  at  A,  Fig.  151.  Take  line  No. 
I,  which  may  be  made  to  half-inch  scale,  and  we 
find  it  will  measure  17  feet  6  inches,  which  is 
required  to  be  divided  into  thirteen  parts.  To 
do  this,  let  AB  on  tongue  of  square  equal  AB  on 
the  right;  make  A5  on  blade  of  square  measure 
6j4  inches,  or  thirteen  halves,  which  are  equal 
to  the  number  of  parts  required.  Draw  line  5B; 
make  5L  equal  one-half  inch;  from  L  square 
over  aline  cutting  in  N;  then  LN  divides  line 
No.  I  into  thirteen  parts,  length  of  each  14''. 

Take  line  No.  2.  This  measures,  by  a  quarter- 
inch  scale,  31  feet  5  inches.  It  is  required  to  be 
divided  into  22  parts.  To  do  this,  let  AC  on 
tongue  of  square  equal  AC  No.  2,  on  the  right, 
and  A4  on  blade  of  square  measure  5>^  inches 
or  22  quarters,  which  is  equal  to  the  number  of 
parts  required.  Draw  line  4C;  make  4R  equal 
%  inch;  from  R  square  over  a  line  cutting  in  K. 
This  gives  RK,  which  will  divide  line  No.  2  into 
22  parts  as  required.     Length  of  each  I's". 

No.  3  on  the  right.    This  line  measures,  by  a 


■     ii 


,1  Ui 


(it* 


I 


196  PRACTICAL  USES  OF 

quarter-inch    scale,    26    feet    6    inches. 


It    is 


""'■■'' ' ' ' 


Fig.  151 


THE  STEEL   SQUARE 


197 


required  to  be  divided  into  17  parts.  To  do 
this,  .aake  AD  on  tongue  of  square  (-qual  AD, 
No.  3  on  the  ri^ht,  and  A3  on  the  blade  of 
square  to  measure  4^  inches  or  17  quarters; 
that  being  the  number  of  parts  required.  Draw 
Hne  3D;  make  3P  equal  %  inch;  from  P  square 
over  a  line  cutting  in  J;  then  PJ  divides  line  No. 
3  into  17  par         Length  of  each  i'7". 

No.  4  on  the  right.  This  line  measures,  by  a 
quarter-inch  scale,  23  feet,  and  it  is  required  to 
be  diy'ded  into  11  parts.  This  is  done  by  ma- 
king AE  on  tongue  of  square  equal  AE,  No.  4  on 
the  right.  Let  A2  on  blade  of  square  measure 
1 1  quarter  inches,  that  being  the  number  of  parts 
required;  draw  line  2E;  make  2F  equal  5<  inch; 
from  F  square  over  a  line  cutting  in  H;  t;2n 
FH  divides  lines  No. 4  into  11  parts.  Length  of 
each  2   Ii't". 

In  dividing  a  space  of  any  great  extent,  the 
quarter-inch  scale  will  be  found  most  convenient. 
To  give  a  practical  illustration:  Suppose  a  line 
lo  1  e  96  feet  long,  and  it  is  requi-ed  t-  divided 
into  48  parts.  Begin  in  a  systemat.^,  way  by 
squaring  over  a  line  on  the  surface  of  a  board, 
\nd  from  its  edge  mark  one  foot  on  the  line; 
one  foot  being  equal  to  48  inches.  Next,  meas- 
ure two  feet  from  the  line  along  the  edge  of 


I'm 
^  if 

r       :'t\ 

1  :i 


:  Sj 


i--*l|| 


198 


PRACTICAL  USES  OF 


board,  two  feet  beiig  equal  to  96  quarter  inches. 
Now  proceed  to  find  one  part  that  wiil  divide  96 
feet  into  48  parts.  The  answer  is  given  by  the 
method  just  explained. 

A  little  thought  will  enable  the  «tudent  to  use 
any  scale  that  is  divisible  by  the  divisions  and 
subdivisions  laid  down  on  the  steel  square,  a 
matter  'ihat  will  enable  him  to  divide  lines  of 
almost  any  length. 

When  we  know  the  side  of  a  square  we*  cm 
readily  find  its  diagonal  by  the  steel  square  as 
follows:  Suppose  AB,  Fig.  152,  be  the  side  of  a 
square  which  measures,  by  a  quarter-inch  scale, 
6  feet  10  inches.  To  find  its  diagonal,  draw  the 
line  BC  to  the  angle  45°;  take  A  as  center  and 
strike  an  arc  touching  line  BC,  cutting  in  V. 
Join  V,  C.  This  gives  bevel  W;  let  it  be  applied 
to  the  square,  Fig.  153,  and  at  t!  e  distance  AB, 
which  is  equal  to  the  side  of  given  square,  lay  a 
straight-edge  against  the  blade  of  bevei  and  the 
line  made  by  it  cuts  in  mark  C  on  the  square, 
giving  AC  for  the  diagonal,  which  measures  g 
feet  7  inches.  This  agrees  exactly  with  line  PR 
or  BC,  Fig.  152. 

Now  let  it  be  required  to  give  the  diagonal  of 
a  square,  the  sides  of  which  are  equal  to  A2,  and 
measure  10  feet  6  inches;   find  its  diagonal  at 


[.-,4.7..- .AM<^.\  •A*it,^i..r,-^U. 


^SSi 


«P1 


THE  STEEL  SQUARE 


199 


Fig-  153  by  making  A2  e-ual  the  side  of  square; 
let  bevel  W  and  the  straigtit-edge  be  applied  as 


before;  then  the  line  from  2  cuts  mark  J  on  the 
square,  giving  AJ  for  the  diagonal,  which  maas- 


r.  ^1 


•  i 

i 


aoo 


PRACTICAL  USES  OP 


11  res   14  feet  8  inches;    tliis  agrees  with  diagonal 
2-3.  I^'ig-  152. 


Fig.  153 

Similar  results  may  be  obtained  without  any 
drawing  by  merely  finding  the  numbers  on  the 
blade  and  ton^^ue  of  a  squa    :  that  will  agree  cr 


' -1- -th^-ai'"" 4~  Y '^: --^.4-tt:'  ^J.-tc^ 


THE  STEEL  SQUARE 


SOI 


equal  the  angle  in  bevel  W,  then  let  the  bevel 
as  now  set  be  applied  to  the  square,  and  we  find 
that  the  blade  of  bevel  agrees  with  mark  4"^" 
on  tongue  of  square  and  6"^"  on  the  blade;  so 
that  the  diagonal  of  any  square  being  required, 
it  is  easily  obtain  d  by  setting  a  bevel  in  the 
manner  staged.  The  answer  will  be  correct  by 
the  angle  in  the  bevel  being  accurate. 


Fig.  154  shows  a  construction  which  might  be 
called  an  attempt  at  squaring  the   circle.      Its 


',.  I 


flOii 


PRACTICAL  USES  OP 


solution  has  been  thought  an  impossibility,  and 
with  all  due  regard  to  the  opinion  of  others  on 
this  point,  we  think  it  quite  possible  to  solve 
this  diflficult  problem  by  a  new  and  simple 
method  of  construction,  which  is  here  given. 

The  diameter  of  the  circle  measures  12  feet. 
From  center  B  draw  line  BD  at  right  angles 
with  the  diameter;  draw  from  B  and  D,  with  the 
angle  45°,  intersecting  in  F;  through  F  draw  3E 
parallel  with  BD;  draw  through  E  parallel  with 
the  diameter,  and  from  center  B  draw  parallel 
with  FB  cutting  line  from  E  in  L;  and  from  L 
draw  parallel  with  BD;  also  from  E  draw  through 
center  B,  cutting  in  2;  draw  through  2  parallel 
with  the  diameter  cutting  line  from  L  in  H,  and 
we  have  now  three  sides  of  a  square;  the  fourth, 
being  made  equal  to  one  of  these,  completes  a 
square,  the  area  of  which  will  be  found  equal  to 
that  of  the  circle;  its  diameter  beinp  12  feet,  and 
one  side  of  the  square  10  feet  5  inches. 

To  remove  all  doubt  as  to  the  correctness  of 
this  solution,  let  us  prove  it  in  another  way,  by 
a  right  angle,  or  the  steel  square,  shown  at 
Fig.  155.  Here  make  the  distance  AC  equal 
the  diameter  of  the  circle;  lay  a  straight-edge 
across  the  square,  keeping  its  edge  on  point  C. 
Take  bevel  K  in  the  angle  HAC  and  apply  it 


THE  STEEL  SgUARE 


303 


to  point  C;  bring  the  straight-edge  against  the 
blade  of  bevel,  and  we  find  a  line  cutting  the 


Fig.  155 


right  angle  at  point  H,  giving  the  distance  HA, 
which  exactly  equals  one  side  of  the  square  HL, 
Fig.  154,  thus  giving  the  same  result  by  two 
different  methods. 


q 

n 


I       r 


PRACTICAL  USLo  OF 

The  utility  of  the  stetl  square  is  now  evident, 
by  it  the  measurement  of  any  surface  may  be 
instantly  given,  and  by  the  same  means  we  can 
find  the  capacity  of  anythinpr  round  or  square. 
All  that  is  necessary  is  to  set  the  bevel  to  a  certain 
number  of  parts  or  inches  on  the  blade  and  tongue 
of  the  square.  To  explain  this  point,  take  bevel 
K  as  now  set,  and  apply  its  stock  against  the 
blade  of  square;  move  the  bevel  until  the  blade 
cuts  some  members  on  the  square  that  will  cor- 
respond with  the  angle  of  the  bevel ;  its  blade 
agrees  with  marks  s"'A"  and  :,%",  or  6"^"  and 
5"^";  either  of  these  numbers  will  answer.  The 
bevel  being  set  in  the  manner  stated,  will  not 
require  any  alteration,  let  the  diameter  of  the 
circle  be  what  it  may. 

In  a  previous  diagram  I  explained  how  this 
problem  might  be  worked  out  by  a  different 
method.  Both  are  correct,  and  the  reader  may 
adopt  either  one  or  the  other. 

I  will  now  endeavor  to  show  how  the  square 
may  be  used  in  getting  certain  dimensions  with- 
out much  effort.  Suppose  we  wish  to  find  the 
superficial  contents  of  a  board  or  other  material 
that  is  not  more  than  one  inch  thick,  we  proceed 
as  follows:  Let  Fig.  156  represent  the  square. 
The  blade  and  tongue  may  be  divided  into  any 


THE  STEEL  SgUARE  ,05 

number  of  parts  by  scale.     V  -  may  call  a  ,',, 
or  an  >i  or  a  >i,  etc.,  a  foot,  just  as  we  wish  to 


u 

-R 

>l 

(> 

i  i* 

It 

'f' 

ti 

V 

i 

*       ..      ". : 


^-il  I  ■   I   I   "'1    I  I   I   I    I  I   I  I   I  I 


fa 


JJI 


•^?^ 


M 


1*11 


Fig.  156 


,■; 


mi 


3o6 


PRACTICAL  USES  OF 


I: 


J' 


meet  the  condition.  Let  the  point  12  or  6.  where 
the  lines  most  converge,  be  a  fixed  point.  Now 
assume  a  board  to  be  26  feet  long  and  6  inches 
wide.  We  know  that  the  surface  measurement 
of  it  is  exactly  12  feet.  The  same  result  is  given 
by  the  right  angle.  For  example,  draw  the  line 
12-26,  and  parallel  to  it  draw  from  the  6-inch 
mark.  The  line  cuts  in  13  feet,  which  is  the 
answer. 

Find  the  measurement  of  a  board  21  feet  long 
and  19  inches  wide.  Draw  the  line  12-21,  and 
parallel  to  it  draw  from  the  19-inch  mark.  The 
line  cuts  in  33^^  feet,  which  is  the  answer. 

Give  the  measurement  of  a  board  18  feet  long 
and  io>^  inches  wide.  Draw  the  line  12-18,  and 
parallel  to  it  draw  the  loK-inch  mark.  The  line 
cuts  in  ist'V  feet,  which  is  the  answer. 

Find  the  measurement  of  a  board  18  feet  long 
and  9  inches  wide.  The  line  12-18  being  already 
given,  draw  parallel  to  it  from  the  9-inch  mark; 
and  this  line  cuts  in  13  feet  6  inches,  which  is  the 
answer. 

What  is  the  surface  measurement  of  a  board 
29  feet  long  and  4  inches  wide?  Draw  the  line 
12-29,  and  parallel  to  it  draw  from  the  4-inch 
mark.  The  line  cuts  in  10  feet,  which  is  the 
answer. 


THE  STEEL   SQUARE 


907 


In  the  foregoing  it  will  be  seen  that  the  scale 
used  is  less  than  one  inch;  but  whatever  the 
scale,  it  must  be  made  to  represent  an  inch  in 
the  end;  thus,  if  we  use  }^  inch,  then  we  must 
multiply  the  result  by  2  to  make  it  into  inches, 
and  if  we  use  a  quarter-inch  scale,  then  multiply 
by  4,  and  so  on,  in  order  to  make  the  result  into 
feet  and  inches. 

To  reduce  surface  measure  to  square  yards  by 
aid  of  the  square,  we  proceed  as  follows:    There 
are  9  square  feet  in  i  square  yard,  as  shown  in 
Fig- 157.  so  we  make  9  a  constant 
number  in  this  problem.   Let  us 
suppose  any  of  the  regular  di- 
visions of  the  square  i  yard  in 
length;   be  it  a  ^-inch,  i-inch, 
or  any  other  division.    To  desig- 
nate the  sides,  call  the  perpen- 
dicular biadc  of  square,  and  lower  line  tongue  of 
square,  and  A  the  internal  angle.     Let  9B,  Fig. 
158,  be  the  fixed  point,  and  from  it  draw  the 
perpendicular,  which  divide  into  any  number  of 
parts,  each  to  equal  those  on  the  right  angle. 

It  is  now  required  to  give  the  number  of  square 
yards  in  a  floor  22  feet  long  and  15  wide.  Draw 
the  line  9-22-H,  and  parallel  to  it  draw  from  mark 
155  cutting  in  H.    This  gives  14?^,  to  which  add 


Fig.  15  7 


-ill 


■■ 


M  I! 


iii 


!■ 


f; 


J 


11 


3o8 


PRACTICAL  USES  OF 


22,  making  365^  square  yards  of  floor,  which  is 
the  answer. 

How  many  square  yards  of  carpet  will  cover  a 
floor  18  feet  long  and  13  wide?  Draw  the  line 
9-18-L,  and  parallel  to  it  draw  from  mark  13-D 
cutting  in  F.  This  gives  8,  to  which  add  18, 
making  26  square  yards  of  carpet,  which  is  the 
answer. 

To  find  the  number  of  square  yards  in  a  floor 
which  is  15  feet  long  and  11  wide.  Draw  the 
line  9-15-K,  and  parallel  to  it  draw  from  mark 
ii-C  cutting  N.  This  gives  3K,  to  which  add 
15,  making  18^  square  yards  in  the  floor;  this  is 
the  answer. 

Now  give  the  number  of  square  yards  on  the 
surface  ot  a  counter  which  is  10  feet  long  and  75^. 
wide.  ,  Draw  the  line  9-10-R,  and  parallel  to  it, 
draw  from  mark  7K-P  cutting  in  J.  This  gives 
8}i  square  yards  on  the  surface  of  counter,  umch 
is  the  answer. 

It  is  now  evident  that  the  steel  square  may  be 
made  to  give  many  other  useful  and  practical 
ideas,  besides  those  which  have  been  shown. 

THE    SQUARE    IN    IIANDRAILING 

For  over  25  years  I  have  felt  certain  that 
some  genius  will  arise  and  show  the  world  how 


THE  STEEL  SQUARE  ,09 

all  kinds  of  handrailing  may  be  "laid  out"  by  the 
use  of  the  steel  square  alone.  I  have  wrestled 
with  the  subject  often  and  long,  but  it  has  so  far 
eluded  me,  though  in  a  hazy  way  I  have  been 
able  to  get  a  glimpse  of  the  relation  between  the 
square,  the  rise  and  tread,  and  an  oblique  cut 
cylinder.  I  know  that  a  relation  exists,  and  that 
relation  and  its  perfect  rendering  will  be  discov- 
ered some  of  these  days  by  a  steel  square  expert; 
and  a  fortune  awaits  the  man  who  makes  the 
discovery  and  gives  it  to  the  public. 

This  may  seem  heretical  to  the  old  time  hand- 
railer  who  has  waded  through  the  mazy  paths 
as  marked  down  by  the  old  master-hands  of  the 
science,  and  they  may  well  be  forgiven  if  they 
turn  up  their  scientific  noses  at  what  I  have  said 
in  this  matter,  and  sneeringly  call  it  so  much 
"bosh."     If  thirty  years  ago  any  person  had  pre- 
dicted that  the  steel  square  could  be  made  to 
accomplish  what  it  now  can  in  good  hands,  the 
prophet  would  have  been  s..t  down  as  a  foolish 
fellow,  and  his  predictions  "all  bosh."     Yet  we 
see  what  has  been  done,  and  knowing  what  I 
do  regarding  the  capabilities  of  the  square,  I  do 
not  hesitate  for  a  moment  regarding  my  reputa- 
tion as  a  prophet  when  predicting  that  all  circu- 
lar   and   elliptical    handrails  will    be    laid    out 


miL 


Ij 


i 


no 


PRACTICAL  USES  OF 


altogether  by  aid  of  the  steel  square  and  a  piece 
of  string  before  the  end  of  the  first  quarter  of 


FiQ.  158 


^^ 


ir 


THE  STEEL  SQUARE 


211 


the  twentieth  century.  With  this  certainty 
before  me,  I  urge  all  young  men,  and  old  ones 
too,  to  sometimes  try  and  find  the  method  I  refer 
to.  That  it  exists  in  the  unseen  land  I  am  as 
confident  as  I  am  of  penning  these  lines,  and  to 
the  man  who  makes  the  discovery  fame  and 
wealth  will  be  the  reward. 

As  an  item  in  this  direction,  I  give  the  follow- 
ing, which  is  by  Mr.  Penrose  of  England,  and 
which  was  sent  me  for  publication  in  this  work. 
I  had  reached  this  point  myself  some  years  ago, 
but  described  it  a  little  differently;  but  think  on 
the  whole,  Mr.  Penrose's  way  of  putting  it  is 
perhaps  better  than  mine,  so  give  it  as  it  came  to 
me. 

In  getting  out  face  molds  it  has  generally 
been  considered  necessary  first  to  unfold  the 
tangents  ind  get  the  heights,  and  by  construc- 
tion get  bevels.  This  methoc'  is  somewhat 
clifferen  ough  results  are  the  same,  but  are 
produced  more  expeditiously, — a  steel  square,  a 
pencil,  and  a  pair  of  compasses  being  used. 
Take,  for  illustration,  a  side  wreath  mitered  into 
the  newel  cap.  The  distance  the  newel  stands 
out  of  line  with  the  straight  rail  is  usually  gov- 
erned by  the  width  of  the  hall,  but  where  there 
is  plenty  of  room  it  is  a  matter  of  taste.    The 


fi 


1 


'\<- 


i  I    * 
>  1 


(I 


lii 


I* 

k 

■:t1l 


1    S  Sii  - 

li  «  ' 


aia 


PRACTICAL  USES   OP 


distance  the  easement  runs  back  is  also  a  matter 
of  cl  jke.  The  method  will  apply  no  matter 
where  the  newel  is  placed,  or  whether  the  ease- 
ment is  less  or  more  than  the  one  step  of  tlie 
example  illustrated.  What  is  meant  by  one  step 
is,  that  the  tangent  of  the  straight  rail  continues 


< 


\v.--- 


Fig.  159 
to  the  point  2,  Fig.  159.    The  tanjrent  2-1  is  level. 
To    produce    the    face    mold,    lay    the    steel 

square  in  the  position  in- 
dicated by  the  lines  i,  2, 
3, 4,  not  the  figure  on  the 
square  at  the  points  num- 
bered, and  transfer  them 
Fig.  160  to  a  piece  of  thin  stuff, 

Fig.  160.  Line  3-4  in  Fig.  160  is  indefinite.  Now 
take  the  length  of  the  long  edge  of  the  pitch 
board  in  the  compasses,  and  with  point  2,  Pig.  iTx). 
as  a  center,  cut  the  line  3-4  in  4  and  draw  :-4. 


THE  STEEL  SQUARE 


»t$ 


Now  1-2  is  the  level,  and  2-4  is  the  pitch  tangent 
on  the  face  mold. 

To  get  the  bevels  and  width  of  the  face  mold 
at  both  ends,  take  the  distance  34  on  the  blade 
of  the  square,  and  the  height  of  a  riser  on  the 
tongue  of  the  squa'-e,  apply  to  the  edge  of  a 
board  and  mark  by  the  tongue;  this  gives  the 
bevel  for  the  lower  end  of  the  wreath.  Mark 
the  width  of  the  rail  on  the  bevel;  this  gives  the 
width  of  the  mold  at  the  lower  end. 

Next  take  the  distance  4-x  on  the  blade  of 
the  square,  and  the  distance  shown  on  the  pitch 
board  by  the  line  squared  from  its  top  edge  to 
the  corner,  on  the  tongue  of  the  square;  apply  to 
the  edge  of  a  board  and  mark  by  the  tongue; 
this  gives  the  bevel  for  the  op  end  of  the  wreath. 
Mark  the  width  of  the  rail  on  the  bevel,  and  this 
gives  the  width  of  the  mold  at  the  top  end. 
An  allowance  of  6  inches  is  made  at  the  lop  end 
to  joint  to  the  straight  rail,  and  2  inches  at  the 
bottom  end  to  form  the  miter  into  the  newel  cap. 
The  springing  line  is  taken  from  the  pitch  board. 

Fig-  159.  in  which  are  shown  the  bevels  and 
the  pitch  board  will  help  to  make  clear  the 
method  used.  The  bevel  at  the  back  of  the 
pitch  board  is  for  the  bottom  end  of  the  wreath. 
The  triangle  has  for  its  base  the  line  3-4,  and  for 


!v! 


\l^ 


I  <- 

ill 


314 


PRACTICAL  USES  OP 


its  height  one  riser.  The  hypothenuse  is  the 
length  of  3-4,  Fig.  i6o,  and  Fig.  i6o  stands  ovor 
Fig.  159.  level  on  the  line  1-2-3,  and  inclined 
from  it  in  this  cast  at  an  angle  of  nearly  45°. 

The  top  end  bevel  is  shown  below  the  pitch 
board.  The  angle  has  for  its  base  the  distance 
4-x,  and  for  its  height  not  one  riser  but  the 
length  of  a  line,  from  the  corner  of  the  pitch 
board  squared  from  its  top  edge.  This  bevel 
will  be  understood  better  by  placing  the  pitch 
board  on  the  line  2-4  and  applying  the  small  tri- 
angle to  it  with  its  base  on  the  line  4-x,  and  its 
point  even  with  the  top  edge  of  the  pitch  board. 
It  will  then  be  at  right  angles  to  the  top  edge  of 
the  pitch  board. 

In  practice,  a  parallel  mold  is  generally  used, 
and  the  wreath  piece  is  cut  out;  both  thickness 
of  plank  and  width  of  molding  being  equal  to 
the  diameter  of  a  circle  that  will  contain  a  sec- 
tion of  finished  rail. 

This  is  a  good  beginning,  and  if  this  much  c^n 
be  accomplished  by  the  square,  why  not  more  on 
the  same  lines? 

In  the  eariier  part  of  this  work  I  have  shown 
how  the  square  may  be  employed  in  laying  out 
strings  for  stairs,  step  ladders  and  similar  work, 
and  if  a  method  or  system  for  setting  out  hand- 


m.' 


THE  STEEL  SQUARE 


ai$ 


rails  for  circular  and  elliptical  stairs  by  the 
square  can  be  evolved,  then  nearly  the  whole 
science  of  carpentry  and  joinery  may  be  devel- 
oped and  explained  by  aid  of  that  wonderful 
instrument,  the  American  steel  square. 


h 
nn 


, 


^i^m 


^E-  I 


"6  PKACTICAL  UiJES  OF 


TABLES 

In  the  following  tables  are  all  the  cuts  necessary 
for  obtaining  the  proper  bevels  to  cut  common 
rafters,  hips,  jacks,  valleys  and  purlins,  either  by 
degrees  or  by  the  use  of  the  steel  square,  for  six 
different  pitches,  namely,  quarter-pitch,  one-third 
pitch,  three-quarter  pitch,  half-pitch.  one  and  a 
quarter-pitch,  and  one  and  a  half-pitch. 

The  figures  to  be  employed  on  the  iquare  to 
get  the  proper  bevels,  are  in  the  last  column, 
right  hand  side  of  the  tables. 

TABLE  I 

QUARTER-PITCH    ROOF 

«7  Degrees,  or  6.inch  Rise  to  T2-lnch  Ron 


Descbiption. 


Common  Rafter. 

Hips 

Jacks 

Valleys 

Purlins  VerMcai . 

Purlins  to  Plane 

of  Roof 


I-  i !  1 


12x6 
12x4i 
12X6 
12X4^ 


nm 


27 
18} 

7 
18J 


60 

71J 

60 

71i 

46 

65) 


Nat 

III 


27 
18} 
42 
18} 
90 
184 


^  3  U 

£13  a 


12X6 

12X4J 

12X10} 

12x4r 

Square 

13X13 


a  1! 


n 


Sq.  of  12x6 

"  :2x4i 
"    i.^xe* 

"  12X4J 
"  12X'2 
"      12Xdi 


THE  STEEL  byUARE 


«I7 


TABLE  a 

ONE-THIRD   PITCH   ROOr 

34  Degrees,  or  8-inch  Rise  to  la-lnch  Rijn 


Dehckiption. 


Common  Rafter 

Hips 

Jacks  

Valleys 

Purlins  Vertical 

Purlins  to  Plane 

of  Roof 


12X8 

12x5j 

13-  10 

12X5J 

Square 

12X14J 


S<|.  of  12x8 

"      12X55 

"      12X8 

"      12x5i 

"      12X12 

"      12x«i 


TABLE  3 

THREE-QUARTER    I'lTCH    ROOF 
37  Degrees,  or  9-inch  Rise  to  12-inch  Run 


Description. 


Conmion  Rafter. 

Hips 

Jacks  

Valleys 

Purlins  Vertical. 
Purlins  to  Plane 
of  Roof 


12X9 
12X6J 
12X9 
12x6i 


i-  -  ^ 


37 

P 
ot 

071 


53 

62} 

53 

62| 

45 

58\ 


X2Z 


37 
27} 
39 
27} 
90 
130i 


12X9 

12X6} 

12X9^ 

12x6} 

Square 

12X141 


Sq.  of  12X9 

"      12X6} 

"      12x9 

"      12X6} 

'•      12X12 

"      12X7J 


tiS  PRACTICAL  USES  OF 

TABLE  4 

ONE-HALF   PITCH    ROOF 
45  Degreei,  or  1 3-inch  Rise  to  isMtiuh  Run 


DnCRIFTIliN. 


Common  Rafter . 

Hipe 

Jacks  

Valleys 

Purlins  Vertical . 

Purlins  to  Plane 

of  Roof 


3, 


S5     Si  >Si 


12X12 
12X8^ 
12X12 
22x8i 


Art    in 

Mi  55 


45 

34  i 


45 
55 
45 
64 


8q.  of  12x12 

"  12X8i 
12X 13 
12X8J 
1»X  12 

12+8i 


TABLE  5 

ONE   AND   ONE-THIRD    PITCH   ROOF 
S4  Degrees,  or  i6-inch  Rise  to  la-inch  Run 


DCSCRIPTION. 


Common  Rafter. 

Hips 

Jacks 

Valleys 

Purlins  Vertical . , 

Purlins  to  Plane 

of  Roof 


J3  i 


12X16 
12X11? 
12X16 
12X111 


a 


54 
44 
54 

44 


♦J  t-«  H 
t->u  DC 


86 
44 
36 
44 
45 
51 


O  w  f''  I      P  *j  y: 

N  3  S"        N  3  » 


rsi 


54 
44 
32 
44 
90 
132t 


12X16 

12xllf 

12x7i 

12Xll| 

Square 

12Xl9i 


Sq.  of  12xl») 

"      12X113 

"      12xlt'' 

'•      12X11? 

"      12X1J 

"      12X»3 


THE  STEEL  SyUAKE 


319 


TABLE  6 

ONE   AND   ONK-IIAl.K    I'lTCH    ROOF 
57  Degrees,  or  18  inch  Rise  to  13-inch  Run 


Dehcbiptjun. 


Common  Itafter  . 

HipM 

Jacks 

Vrtlleys 

Purlins  Vertical  . 

Purlins  to  Pliine 

of  Roof 


fl     |-Si 
£5     t^ 


12X18 

i2xr.J» 

1','X1M 
12X121 


•a  t 


33 
44 
iJ3 
44 
45 
50 


m 

i2^ 


57 

46 
30 
46 
ttO 
120 


III 


12X18 

12X12« 

12x»i 

12x12* 

Square 

12x21 


£-■3 
>33 


Sq.  of  12X18 

"      12X12| 

"      12X18 

"      12X121 

"      12X12 

"      12x10 


I  have  given  the  pitches  of  those  roofs  which 
are  more  generally  used  than  any  other,  though 
the  same  rules  which  obtained  the  above  figures 
could  be  continued  indefinitely.  It  has  been 
thought,  however,  that  the  foregoing  examples 
were  quite  sufficient  for  all  ordinary  purposes. 

In  these  tables  it  will  be  seen  that  the  bevels 
for  purlin  cuts  have  been  given,  both  when  the 
purlin  is  square  or  plumb  with  the  horizon,  and 
v\hen  it  sits  with  one  of  its  sides  against  the  raft- 
ers, or  inclined  with  the  roof.  The  square  is 
made  to  produce  all  these  bevels. 


ii 


iao 


PRACTICAL  USES  OF 


i    A 


m 

m 

wM 

i  : 

wi-  ^^H' 

I 

t -^ 

ii 

Hi 

'  •■,\ 

f^H^Bj 

FOR    ESTIMATING   CONTENTS    OF    RAFTERS 

In  the  earlier  pages  of  this  work,  I  promised 
to  publish  a  table  wherein  the  contents  of  rafters 
might  be  estimated  without  being  obliged  to  take 
actual  measurements  of  the  timbers,  and  to  this 
end  the  annexed  table.  No.  7,  is  presented. 

In  the  lengths  given,  there  is  no  provision 
made  for  projections  over  eaves,  or  for  ridge 
poles.  The  measurements  are  from  the  face  of 
the  plate  to  the  point  of  ridge.  The  length  of 
any  rafter  is  given  for  roofs  having  a  pitch  of 
one-quarter  to  one-half,  and  a  span  of  from  8 
feet  to  50  feet.  Xo  provision  is  made  for  frac- 
tions of  feet  in  width  of  building. 

The  length  of  the  rafter  being  obtained,  and 
its  sectional  area  being  known,  the  contents  may 
readily  be  found:  Thus, suppose  width  of  build- 
ing to  be  34  feet,  rafters  to  be  2x6  inches,  sec- 
tional area;  pitch  9-inch  rise;  then  we  have 
length  of  rafter  by  table,  is  21  feet  3  inches,  and 
as  each  foot  in  length  of  a  rafter  2x6  inches  con- 
tarns  one  foot  board  measure,  we  have  21  feet  3 
mches  as  the  amount  of  material  in  each  rafter, 
board  measure.  So  with  all  the  other  dimensions 
in  the  table.  The  lengths  of  rafters  are  given. 
Determine  the  sectional  areas  of  rafters,  and  the 
contents  may  easily  be  found: 


THE  STEEL  SQUARE 


921 


w 


.c 
c 

c 

a 

(^  .2 
O  "o 

(3 

M    CIS 

H  ^ 
^  -g 
W  ►^ 
-^  *.- 

a< 


•asm 


jOO'^H        00^        00^        00  TJ< 


OO'^        00T*<        00 


.i: —I  Ti  -<  ^  ^  >-«  o<  Ti  c<  o  I'*  ?»_w  rerjccM  00  ^  • »" 


•asm 


j5D0D0i>-iWTr«f»05O?I'^l0f»XO««C01C»Xi3 
ir 1-1  — <  T~  »—  —  1^  •?»?>/?)  T<  C^  ?<  W  CC  CO  OS  CO  cc  ■^ 


5«50C>»3503  =  t=i.-;,-.OOt-^t-(M»  OT  OS  ^=  lO 

*; —-<>--<  —  .-I  —.?>  o<  gi  Tj  ?)  o>  rr  oa  co  55  to  r? 


So'^foosascoasox? 


J10l>XO-<05-^!C(-3SOCOOSl~tOXS;0(?Je0  1.'5® 
±! "-"  —I  —  —'—'—''?>'?)  0>  ?<  C>  l?>  C)  ?t  05  OS  CO  CO 


•3STH 
U3UI-81 


iJt^OlCCOSX)  —  !C— •'J"3SC»t-  =  OOC0  *  ^®  ^5« 

^Ult-SDOS  —  SJ-*lIS5Cx3ST-.o>tW»XOS»-i(rJC01ffl 
i: "^  ^  -"  -^  —  —  —  0<  IM  i?>  O)  O)  ?!  Ol  CO  03  CO  05 


•ssia 


•3sia 

qouioi 


i:iAcsi-i®oe<;c.-icci» 


"^'  X  *  IC  Ci  -"  O  O  Ol  W  th 


tiU5«e»(»OM05->t»r-05O— COtlCI^XCJ— iNOS 
±t i-*^^— ^■^^^'— — *o^'?^c^  o>  o>  O  7>  0»  CO  05  CO 


;  -*?>—»:^^r^ 


-S-JC— i.l.,x  — 


i;o^®Or-ii.'sxscot-.-iN«co-Htcxo'j'i-»-cj« 
^rt^t-wo^os-fo^xaso^cs-^^cr-xa  —  e} 

-^i; *^  ^^  '^  —  ^^  ^*  "^  -^  C>  ?>  ??  Ci  gf  o  OJ  CJ  05  05 


CO  »  3S 


^  o  to  I-  X  »  —  ^  CO  i-i  50  r-  00  o  -^  ?>  CO  i."  «  r-  X  o  t— 

.^^ — '  —  —  ~  —  -^  -^  ^  gJ  ?'  <M  OJ  ?>  gt  Tl  ?)  CO  CO 


— ;ji-x— f  r?«t  — »■— M 


■^sjji    —  *  ®  '^  =='-  =5  ooj  Lo  t-  o  3  01 1.0  »-  o  d  Mt-  t-  o  5 

UOUT-H  I  '^  *"* 

*"     »:'*«cj>xosoo}os-towxs5C—  o»TCLO-jr-x© 
" r'.;^-'^  "  — '  ^  —  '-'  o>  o>  o  o»  o<  o.(  g»  o)  CO 


•3stM 


O  t-  »  O  ioj  ^- -J  X  O        i-COlCJ'Ol^ 
^■*l.0»X05O—  OJCO'OtCI-XSiSOlCO-tlCSOt^X 

-^^ '-'"'-'  —  —  —  —  '-"-  o«  oj  e<  c(  oj  o}  g»  (» 


•■>sra 


JounH 


e 
is 

H 

a 
ij 

o 

H 
H 


i:  o  t-  X  o  --  =  oj  CO  1.0  '-r  (-  3s  «      —  o»  -qTiff  i-  x  o  >- 
j'*i-o:c(-x©-<o)co-^i.o:ct-c:©  —  e»05-«fo«i> 

.^^ "—'—'—'  —  —'  —  —  -•  01  0>  0»  CT  0<  C«  0<  P< 


X  ©  oj  -r  ffi  X  o  o 

r-  >-«  .-1  «  —I  01 


sS^^^^SSig 


I  I 


XR/W*MnBf 


rn.  I 


i\ 


if 


PRACTICAL  TREATISE  ON  THE  STEEL 
SQUARE. 

VOL.  II. 

The  student  will  be  expected  to  read  carefully 
these  papers  before  doing  any  work.  His  name 
and  address  will  recjuire  to  be  given  on  each  page. 
He  will  be  expected  to  write  up  the  (jucstions  in 
a  neat  and  intelligent  manner,  using  his  own  style 
and  language,  representing  the  answers  in  such  a 
manner  as  will  be  intelligible — make  all  drawings 
as  clear  as  possible,  and  wherever  they  can  be 
lone  render  them  in  India  ink.  Let  each  answer 
be  original,  do  not  copy  either  from  the  instruc- 
tion paper  nor  from  any  other  source.  The  paper 
used  may  be  of  any  kind,  provided  that  it  is  clean 
and  durable.  Do  not  attempt  an  answer  until 
you  have  thoroughly  grasped  the  subject. 


QUESTIONS. 

1.  Give  description  of  "steel  square"  from 
Augusta,  Maine. 

2.  Give  some  examples  of  how  this  square 
may  be  used. 

3.  Give  description  of  color  and  coating  of  sev- 
eral steel  squares,  and  what  may  be  termed  "the 
best." 

222 


i^^ 


If 


THE  STEEL  SCJUARE 


223 


4.  Give  description  and  sketch  how  to  find  by 
use  of  square,  the  cuts  for  each  side  of  a  roof  hav- 
ing- a  box  12  inches  square  set  on  the  ridge  of 
roof. 

5.  Show  by  sketch  and  descril)e  how  to  deter- 
mine by  a  steel  square  tlie  result  of  any  number 
—for  example,  6  multiplied  by  the  sine  45°. 

6.  When  the  rise  and  seat  of  the  corner  posts 
are  given  of  any  rectangular  framework  wiiich 
slopes  alike  on  all  four  sides,  show  how  to  find 
the  cuts  for  the  ends  of  tlie  corner  posts,  and  the 
blade  of  the  bevel  to  be  applied  to  the  two  faces 
adjoining  to  the  ridge  line. 

7.  When  the  run  and  rise  of  a  common  rtjof 
are  given,  and  we  wish  to  place  upon  it  a  perpen- 
dicular square  pipe  to  stand  upon  the  roof  dia- 
mond-shaped, show  how  by  the  steel  hf|uare  to 
find  the  cuts  for  the  bottom  end  of  tlie  pipe,  and 
the  blade  of  the  bevel  to  be  applied  to  the  two 
faces  adjoining  the  lowermost  vertical  edge. 

8.  Suppose  we  wish  to  cut  an  opening  in  a  roof 
for  a  round  pipe  or  a  tile,  so  that  the  pipe  or  tile 
will  stand  vertical  through  the  opening,  show  by 
sketch  how  the  exact  form  of  the  opening  may  be 
obtained. 

9.  Describe  how  by  use  of  a  steel  square  the 
size  of  a  pulley  that  would  be  required  to  replace 
one  in  use,  if  speed  is  reduced  or  increased. 

10.  When  the  pitch  of  cogs  and  diameter  of  a 
wheel  are  given,  show  how  by  use  of  steel  square 
to  find  the  number  of  cogs  in  wheel. 


?24 


PRACTICAL  'JSES  OF 


l»    'l 


'     2. 


i      J 


11.  When  the  diameter  of  a  circle  is  given, 
sl.^vv  how  to  find  the  side  of  a  square  of  equal 
area,  by  use  of  the  steel  square. 

12.  Show  how  by  use  of  steel  square  to  find 
till'  number  of  square  yards  in  a  given  area. 

13.  Show  how  by  use  of  steel  square,  to  deter- 
mine the  circumference  of  a  circle  when  the  diam- 
eter is  given. 

14.  When  the  area  and  diameter  of  any  circle 
is  known,  show  how  the  area  due  to  any  other 
diameter,  or  a  diameter  due  to  any  other  area, 
may  be   determined. 

lo.  Gi'-e  description  and  sketch  showing  the 
advantage  of  the  steel  square,  in  determining  the 
proper  depth  of  "core-boxes." 

U).  Show  how  to  find  the  diagonal  of  a  sciuare 
or  parallelogram    by  use  of  tl;e  S(iuare. 

1/.  Show  how  to  find  the  circuipference  of  an 
ellipse  or  oval,  by  use  of  the  scpiare. 

18.  Show  hcnv  to  'ind  the  side  of  the  greatest 
square  which  may  be  inscribed  within  a  circle,  by 
use  of  the  square. 

10.  Show  how  to  inscribe  three  small  circles 
within  a  large  circle  of  given  diameter,  set  to  6yj 
inches  on  tongue  and   14  inches  on  blade. 

20.  Show  how  to  get  the  length  of  a  hoop  for 
a  wooden  t;ink  by  the  steel  square. 

21.  Show  "arithmetically"  how  to  find  the 
length  of  hoop  for  same  tank. 

22.  The  proper  angle  for  ordinary  door  and 
window  sills  is  about  one  inch  drop  to  the  foot, 


ii 


THE  STEEL  SQUARE 


225 


show  method  of  finding  this  incHnation  by  use  of 
the  steel  square. 

2i.  The  arms  of  a  straight  horizontal  lever  are 
8  and  12.  A  weight  of  9  lbs.  is  suspended  from 
the  shorter  arm,  what  weight  will  balance  it  on 
the  longer  arm,  show  by  use  of  steel  square. 

24.  Show  how  much  power  is  required  to  sup- 
port a  weight  of  4  lbs  on  an  incline  of  5  in  30,  by 
use  of  the  steel  square. 

25.  A  body  is  weighed  in  a  false  balance  and 
in  one  scale  appears  to  be  9  oz.  and  in  the  other 
12  oz.  What  is  its  true  weight?  Show  this  first- 
ly by  arithmetic  and  secondly  by  use  of  the  steel 
square. 

26.  A  spout  is  20  inches  square,  what  is  the 
diameter  01  a  cylindrical  one  with  the  same  area 
of  L.oss-section?  Show  how  to  obtain  by  use  of 
the  steel  square. 

27.  Show  how  by  use  of  the  steel  square  the 
length  and  angles  of  a  brace  of  irregular  run  or 
any  run,  may  be  obtained. 

28.  Show  by  descriptio-.  and  sketcli  how  to 
give  a  square  stick  an  octagonal  shape  where  frac- 
tions of  an  inch  are  involved. 

29.  Show  how  to  divide  a  board  7  inches  wide 
into  ^our  equal  parts,  by  use  of  "the  rule." 

30.  Show  how  to  square  a  board  by  use  of  "the 

rule." 

31.  Show  how  to  strike  a  circle  and  have  noth- 
ing but  a  rule. 


226 


PRACTICAL  USES  OF 


-i  I 


■<V^ 


I .  I 


32.  Show  by  skctt'i  and  description  how  to 
find  the  distance  across  a  body  of  water,  without 
crossing  same. 

33.  Show  by  sketch  and  description  how  ■\ 
tree  or  any  inaccessible  hei^dit  can  be  measured. 

34.  Show  by  sketch  and  description  how  to  ob- 
tain the  proper  angles  and  bevels  of  trestles. 

35.  Show  by  sketch  the  end  of  a  framed  trestle 
with  posts  on  ihe  same  inclination,  also  the  braces, 
one  framed  in  the  angles  and  the  other  spiked  on. 

36.  Show  by  sketch  and  describe  how  to  get 
the  bevel  cut  on  a  square  stick,  say  4"x4". 

37.  Give  description  how  to  obtain  the  lengths 
and  cuts  of  hip  rafters  on  a  pitch  of  45",  by  the 
use  of  the  steel  square. 

38.  Give  description  and  sketch  showing  how 
to  obtain  the  side  cut  on  hip  or  valley  rafter,  by 
use  of  square. 

39.  Give  plan  of  the  valley  rafter  as  laid  off 
for  cutting  bevels. 

40.  Give  sketch  showing  position  where  ridges 
meet. 

41.  Give  description  and  sketches  showing  the 
valley  rafter  in  its  proper  position  on  the  wall- 
plats,  and  how  to  find  the  hypothenuse  or  line  of 
rafter  supposing  that  the  rise  is  9  inches  to  the 
foot  run,  by  use  of  steel  square. 

42.  Give  description  and  sketch  showing  the 
method  of  obtaining  the  bevel  on  valley  rafter,  by 
use  of  the  square. 


4. 


THE  STEEL  SyUAKE 


227 


43.  Suppose  the  half  width  of  a  roof  ha-ing  a 
pitch  of  45  '  degrees  i«  10  feet,  and  that  an  ad- 
joining roof  -'s  one-third  pitch,  how  many  feet  will 
it  take  to  make  an  equal  rise? 

44.  Describe  how  to  frame  a  rafter  against 
two  ridge  boards  running  at  right  angles. 

45.  When  the  pitches  are  the  same  on  all  sides 
of  a  roof  and  the  building  is  square,  at  what  angle 
of  degrees  do  the  hips  or  valleys  run  in  from  the 
corners  ? 

46.  Give  description  and  sketches  showing 
how  to  find  the  top  cuts  of  left  and  right  jack- 
rafters  on  a  roof  plan  with  the  one-third  pitch 
on  the  main  part,  with  a  half  pitch  gable. 

47.  In  connection  with  an  uneven-pitched  roof, 
where  a  projecting  cornice  is  desired,  with 
plancer,  at  what  point  vill  the  valley  rest,  and 
explain  how  much  the  plate  on  the  steeper  pitch 
will  be  raised  above  that  of  the  lower  pitch. 

48.  Give  sketch  ?nd  description  showing  how 
to  find,  by  use  of  two  "squares,"  the  runs  of  com- 
mon rafters,  runs  of  jack  rafters,  short  valley  and 
long  valley,  where  the  rises  in  the  roof  are  of  dif- 
ferent heights. 

49.  Give  description  and  sketch  showing  how 
to  get  the  bevel  of  the  top  edge  of  the  jack  rafter, 
by  use  of  the  square. 

50.  Give  description  and  sketch  showing  how 
to  i;et  the  bevel  for  the  side  of  the  purlin  to  fit 
against  the  hip  rafter,  by  use  of  the  square. 


>l\        .i   : 


? . 


228 


PRACTICAL  L'SES  OF 


51.  Give  description  aiul  sketch  sho\vin.<i;  how 
to  get  edge  bevel  of  purlin,  by  use  of  the  scjuare. 

52.  Give  description  anci  sketch  showing  how 
to  cut  "jacks"  for  curved  roofs. 

53.  Give  description  and  sketch  showing  how 
to  get  hip  rafters  for  curved  roofs,  without  using 
the  square  for  the  purpose. 

54.  Give  description  and  sketch  showing  how 
to  obtain  the  lateral  curve  of  valley  rafter. 

55.  Give  description  and  sketch  showing  how 
to  get  the  side  bevels  and  lengths  for  jacks  in  a 
hipped  roof. 

56.  Give  description  and  sketch  showing  how 
to  get  bevel  or  backing  of  "hips." 

57.  Give  description  and  sketch  showing  how 
to  lay  out  an  octagon  bay  window. 

58.  Give  description  and  sketch  showing  how 
the  framing  of  an  octagon  roof  of  tower  or  spire 
is  formed  when  it  intersects  another  roof. 

Z9.  Give  description  and  sketch  showing  how 
the  pitch  of  a  tower  roof  may  be  obtained  by  use 
of  the  square. 

60.  Give  description  and  sketch  showing  the 
method  of  laying  oflf  the  lines  for  an  octagonal 
roof  having  curved  rafters. 

61.  Give  description  and  sketch  showing  the 
method  of  obtaining  the  curves  for  hip  and  jack 
rafters  of  an  octagonal  roof. 

62.  Give  sketches  sliowirg  the  plan,  elevation 
and  section  in  the  construction  of  an  octagonal 
dormer  roof. 


ni 


1L' 


THE  STEEL  SQUARE 


229 


63.  Give  sketches  of  three  kinds  of  "hoppers," 
i.  e.,  triangular,  square  and  hexagonal. 

64.  Describe  the  geometrical  problems  in- 
volved in  these  cases. 

65.  Give  description  and  sketches  showing 
how  all  the  lines  and  cuts  may  be  obtained  by  use 
of  the  square,  in  the  construction  of  these  hop- 
pers. 

66.  Give  description  and  sketch  showing  how 
to  find  the  side  cut  and  miter  of  large  hoppers. 

67.  Give  description  and  sketch  showing  an- 
other method  for  obtaining  same,  taken  from  the 
old  "American  Builder." 

68.  Give  description  and  sketch  showing  how 
to  represent  the  elevation  of  a  box,  the  sides  of 
which  have  a  slope  of  45\ 

69.  Give  description  and  sketches  showing 
how  the  miters  and  cuts  of  a  "hopper"  may  be  ob- 
tained by  use  of  the  scjuare. 

70.  Give  description  and  sketch  showing  an- 
other method  of  finding  the  miter,  and  the  angle 
of  co-pitch  standing  at  an  angle  of  90°  with  the 
given  pitch. 

71.  Give  description  and  sketches,  showing 
how  to  obtain  the  bevels  and  cuts  for  nearly  all 
kinds  of  flared  work  or  inclined  framing  as  de- 
scribed by  the  late  Robert  Riddell. 

72.  Give  description  and  sketches  showing 
how  to  lay  out  the  lines  for  every  conceivable  cut 
required  in  tapered  framing,  paneling  or  splayed 


^1  . 


Mi 


H 


!     I' 
1     P 


pi 

If" 


* 


-f  k 
I 


230 


PRACTICAL  USES  OF 


work  of  any  kind  when  angles  are  thrown  out  of 
square  (according  to  nietliod  accredited  to  Peter 
Nicholson  and  improved  by  the  late  Robert  Rid- 
dell). 

73.  Give  description  and  sketch  showing  how 
to  divide  a  line  of  any  length  into  any  number  of 
equal  parts,  by  the  sieel  scpiare. 

74.  What  scale  is  the  most  convenient  in  divid- 
ing a  space  of  any  great  extent,  and  give  an  ex- 
ample? 

75.  Give  description  and  sketch  showing  liow 
by  the  steel  s(|uare  to  find  the  diagonal  of  a  square 
when  one  side  of  the  s<|uare  is  given. 

76.  Give  description  and  sketch  showing  how- 
to  get  the  diagonal  of  a  s(|uare.  the  sides  of  which 
measure  10  feet  6  inches,  by  use  of  the  steel 
sciuare. 

77.  Give  description  and  sketch  that  shows  a 
construction  of  what  may  be  termed  "an  attempt 
at  squaring  the  circle"? 

78.  Give  proof  of  the  correctness  of  this  solu- 
tion by  means  of  the  steel  s(|uare. 

79.  Give  description  of  how  the  utility  of  the 
steel  square  may  be  applied  to  the  measurement 
of  any  surface,  or  to  find  the  capacity  of  anything 
round  or  squan . 

80.  Give  description  and  sketch  showing  how 
to  obtain  by  the  steel  square  the  superficial  con- 
tents of  a  board  or  other  materia!  th.it  is  not  more 
than  one  inch  thick,  and  26  feet  long  and  6  inches 
wide. 


ii 


IC». 


THE  STEEL  SQUARE 


231 


81.  Give  description  how  to  find  solution  of 
same  by  the  right  angle. 

82.  Give  description  liovv  to  find  the  measure- 
ment of  a  board  21  feet  long  and  19  inclics  wide, 
by  steel  square. 

83.  Give  description  how  to  find  the  surface 
measurement  of  a  board  18  feet  long  and  lOy^ 
indies  wide,  by  the  use  of  the  steel  s(|uare. 

84.  Give  description  how  to  find  the  surface 
measurement  of  a  board  18  feet  long  and  9  inches 
wide,  by  use  of  steel  square. 

85.  Give  description  how  to  find  the  surface 
measurement  of  a  board  29  feet  long  and  4  inches 
wi'lc.  by  use  of  the  steel  square. 

86.  Give  description  of  how  to  reduce  surface 
measure  to  scpiare  yards  by  use  of  the  steel  s<|uare. 

87.  Give  description  of  how  to  find  the  num- 
ber of  square  yards  in  a  floor  22  feet  long  and  15 
feet  wide,  by  aid  of  the  steel  s(|uare. 

88.  Give  description  of  how  to  find  the  num- 
ber of  square  yards  of  carjjct  that  will  cover  a 
floor  18  feet  long  and  13  feet  wide,  by  use  of  the 
steel  square. 

89.  Give  description  of  how  to  find  the  num- 
ber of  s(iuare  yar;ls  in  a  floor  which  is  15  feet 
long  and  1 1  feet  wide,  by  use  of  the  steel  square. 

90.  Give  description  of  how  to  find  the  num- 
ber of  square  yards  on  the  surface  of  a  counter 
which  is  10  feet  long  and  7>4  feet  wide,  by  use 
of  the  steel  scjuare. 


11 


^' 

\    H 

J 

t  f  •• 

3  1 

W- 

III" 

i!  ■ 

i 

* 

-  m 

-^  -•  ■          4 ■Hftl ^2 

Ir 

*'- 

:MFJ^^^'r  '»<... 


INDEX  TO  VOLUME  II 


Author's  Preface  3 

Aii'ithrr  Steel  Square 7 

All  Kni,'lish  Mithod  of  L'sitifj  the  Square loo 

An  Octagon  Towir 123 

Anjjit's  and  Cuts  in  Octatjon  Work 126 

A  Mithod  of  Laying  Out  f,"ur\<-d  Rafters 136 

Al!  Mitt  rs  for  Hopjxrs 163 

Another  Method  of  Obtaining  Miters 168 

A  Method  of  Hopixr  Lines 184 

A  rian  of  Lines  for  All  Kinds  of  Hoppers 186 

Acute  Miters  for  Hoppers l86 

B 

Bevels  of  Slopes 28 

He\  tds  of  Window  and  Door  Sills 43 

liay  Windows,  Octagcn 48 

Hacking  of  Hi|)s 115 

Bevels  for  Backing 1 18 

Bay  Windows 12c 

Bevels  for  Hoppers 1 59 

Butt  Cuts  for  Hoppers 1 5g 

Bevels  for  Odd-shaped  Hoppers 162 

Bevels  for  All  Kinds  of  Hoppers 186 


Color  of  Squares 10 

Care  of  Squares  Generally 11 

i 


i         (I 

'■A  •■ 


II 


if 


.'t  i 

:\  ~ 

'  i 


fi'     t 


M  '  r 


ii  INDEX  TO  VOLUME  II 

Crenelated  Squares '^^ 

Cutting  for  a  Round  Pipe  through  'incline.' ".'.'""     H 

Center  Hip  Rafter *  *       ^ 

Concerning  Roof  Framing 1! 

Cutting  Double  Bevels  .. .  f- 

Croker's  Method [[[] 

Cut  of  Right  Jack l^ 

Cut  of  Left  Jack Z^ 

Cuts  for  Jacks ^^ 

Cut  of  Main  Jacks ^ 

Cut  of  Gible  Jacks ^ 

Combination  Diagram,  by  Mr',  'woods.' .' ?? 

Cardboard  Diagram ^^ 

Curved  Rafters  ...  ^^ 

Cove  Rafters ............'. '°^ 

Curved  Octagon  Rafters !?2 

Curved  Rafters  by  Lines [ 

Co-Pitches  and  Other  Pitches'  'for  *Hoppers' .' I77 

Corners  for  Hoppers '  *     i/ 

Cuts  for  Six  Different  Pitches 216 

Contents  of  Rafters  for  Different  Lengths. ." .' '.'..'.  ]  221 

D 

Description  of  Complicated  Squares ... 

Diedral  Angles ^ 

Diagram  for  H ip  Jack  Rafters  ......."' iy 

Double  Curved  Raftcrs-Ogcc " ' ' j , [ 

Diagram  for  Backing  Hips - 

Describing  an  Octagon ^ 

Describing  an  Octagon  Bay  Window Wz 

Diagram  of  Octagon  Tower 

Dormer  Window ^^ 

Dormer  Window  Front  ^"^j 

140 


•■."■■■?  "^SKtiirw-^z 


tMM.9*iHiM%ifF 


INDEX  TO  VOLUME  II  m 

Page 

Dormer  Window  Plan . . . , 147 

Diagram  for  Hoppers ice 

Diagram  of  Hopper  Cuts 160 

Dividing  a  Circle  into  Equal  Parts 193 

Different  Pitches  and  Their  Cuts 216 

E 

Elevation  of  Octagon  Timber  Tower 127 

Elevation— Another  Vf  w 128 

Elevation  of  Dormer  Window 147 

Elevation  of  Hoppers 152 

Elevation  and  Plan  of  Hoppers 170 

F 

Fitting  a  Box  Diagonally  over  a  Ridge 14 

For  Raking  Mouldings  and  Cornices 25 

For  Working  Core  Boxes 34 

P>aming  Octagon  Roofs ng 

Framing  for  Dormer no 

Framing  Octagon  Bays 120 

Framing  Octagon  Tower 126 

Framed  and  Notch  cl  Timbers 133 

Finished  Sketch  of  Octagon  Tower 153 

Flares  of  Hoppers i^g 

Finding  the  Circumference  of  a  Circle 190 

G 

Graphic  Method  of  Finding  Areas  of  Circles 31 

General  Items  by  Stoddard 84 

General  Pitches  of  Six  Kinds 216 

H 

How  to  Obtain  Length  of  Huups  for  Tanks 42 

How  to  Bevel  Window  or  Door  Jambs 43 


b 


I  ^ 


li  - 


'If 


i 


11     ■    I 


iv  INDEX  TO  VOLUME  II 

P&ffC 

Hips  and  Valleys,  by  Stoddard s6 

Henry  Cook's  Method  of  Laying  Out  Roofs 96 

Hick's  Method  of  Roofing 106 

Hoppers  and  Hopper  Bevels 152 

Hopper,  Three-sided je^ 

Hexagonal  Hoppers ,5 . 

Hopper  Cuts  by  the  Steel  Square ] .    161 

Handrailing 2i» 

Hips  for  Six  Pitches 216 

I 

In  Laying  Off  Rafters 53 

In  Laying  Out  Jack  Rafter  Bevels ....    ,,,,  68 

Irregular  Pitches [  -g 

J 

Jack  Rafters ,  j 

Joints  in  Hopper  Work ' .  162 

Joints  For  Hopper  by  Steel  Square 178 

Joints  in  Splayed  Work,  by  Riddell igi 

Joints  for  All  Kinds  of  Hoppers ] . .  186 

Jacks  for  Six  Pitches 216 

L 

Laying  Out  Sizes  for  Pulleys jg 

Laying  Out  Cogs  in  Toothed  Gear ig 

Lines  for  Oblique  Framing 22 

Laying  Off  Hip-Backing .'  ]  53 

Length  and  Bevel  of  Jack  Rafters "  71 

Laying  Off  Valleys 75 

Lengths  of  Jack  Rafters— Another  Method 104 

Length  of  Cripples  by  the  Square 106 

Lines  for  Curved  Rafters ,,n 


Ifi! 


INDEX  TO  VOLUME  II  v 

Page 

Lines  for  Hoppers 154 

Lumber  Measurement  by  the  Square 210 

M 

Multiplication  by  Aid  of  Square 15  | 

Main  Rafters 50 

Measuring  Inaccessible  Distances 56 

Making  Trestles 60 

Method  of  Laying  Out  Timber  Octagon  Tower. . .  130 

Miter  Cuts  for  Hoppers 158 

Method  of  Getting  Joints   for  Hoppers,    by  Mr. 

W'  ods 172 

Miters  for  Obtuse  and  Acute  Angle  Hoppers 186 

Measurements  by  the  Square 207 

N 
Nicholson's  Method  of  Hip  Roofing jg 

O 

Obtuse  and  Acute  Angles 30 

Octagon  Bay  Windows 48 

Ogee  Rafters 107 

Octagon  Framing 120 

Octagon  Bays 121 

Obtuse  and  Acute  Cuts  for  Hoppers i86 

P 

Polygons  of  All  Kinds 20 

Proportional  Reduction  of  Mouldings 39 

Proportioninjj  Spouts 45 

Practical  Use  of  Square  and  Rule 52 

Pope's  Method  of  Roofing 78 


:.  K 


^  I 


N ,  n 


vi  INDEX  TO  VOLUME  II 

Pitches  of  Roofs ^q 

Positions  of  Hips  and  Valleys qi 

Plan  of  Octagon  Roofs 124 

Pitches  and  Scales  for  Towers  134 

Perspective  View  of  Framed  Dormer 145 

Plumb  Cuts  for  Rafters 13S 

Plan -of  Timber  Work  for  Dormer  Window 147 

Plan  of  Octagon  Tower 142 

Pitches  of  Octagon  Tower 143 

Plan  of  Dormer  Base 140 

Plans  of  Hoppers 156 

Planceer  Cuts  for  Cornice im 

Plan  and  Elevation  of  Hopper 170 

Pitch  Line  for  Hoppers  j  -_, 

Problems  in  Handrailing  by  the  Square 212 

Pitches  of  Various  Kinds 216 

Purlin  Cuts  for  Six  Pitches 217 

Q 

Quick  Methods  of  Laying  Out  Octagon  Sticks. . .     46 

Quick  Methods  of  Obtaining  Hopper  Cuts 163 

Queries  in  Hopper  Building lyg 

R 

Rules  for  Inclined  Framing 23 

Roof  Framing co 

Run  and  Rise  of  Rafters 6; 

Run  and  Rise  of  Jack  Rafters -q 

Run  of  Hips j^- 

Run  of  Val  leys gj^ 

Rafter  Patterns 104 

Riddel I's  Methods  for  Hopper  Work 182 


INDEX  TO  VOLUME  II  ▼« 

Page 

Remarks  on  Handrailing 214 

Ratter  Tables,  Pitches,  etc 217 

S 

Some  Odd  Problems 13 

Speeding  Pulleys 18 

Some  Good  Things 49 

Side  Cuts  for  Valley  Rafters 72 

Stoddard's  Method  of  Roofing 81 

Spacing  Off  a  Rafter 85 

Short  Jacks 88 

Sections  of  Hips  and  Valley  Rafters 94 

Smith's  Improved  Method  of  Roofing 112 

Scale  Elevations  of  Pitches 134 

Steep  Pitches,  and  How  to  Work  Them 135 

Side  Elevation  of  Dormer 148 

Skeleton  Frame  of  Tower 150 

Square  Hoppers 153 

Some  Hopper  Lines 175 

Some  Remarks  on  Hopper  Work 180 

Stair  Railing 213 

T 

To  Find  Area  of  Given  Circle 20 

To  Find  Number  of  Yards  in  Given  Area 21 

To  Inscribe  Polygons  within  Circles 36 

To  Find  the  Apothems  of  Polygons 38 

To  Obtain  Length  of  Hoop  for  Barrel 54 

To  Measure  across  a  River 58 

To  Measure  the  Height  of  a  Standing  Tree 59 

Timber  Framing  in  Octagon  Tower 126 

Triangular  Hoppers 1 52 

The  New  Hopper  Lines 177 


i 

1 


^.1: 


viii  INDEX  TO  VOLUME  II 

T'le  Square  as  a  Calculating  Machine 199 

Tables  for  Rafters  216 

To  Find  Cubical  Contents  of  Rafters 221 

U 

Unequal  Pitches 76 

Uneven  Pitches 80 

Uneven  Valleys  114 

V 

Valley  Rafter  Bevels 64 

Valley  Rafter  and  Cripple  Cuts 73 

Valley  for  Uneven  Pitched  Roof 1 14 

Valleys  for  Six  Pitches 216 

VV 

Wood's  Method  for  Hips  and  Valleys 83 

Work  on  Cornices i6g 

Wood's  Method  of  Working  Hoppers 176 

Review  Questions    222 


HOUSE    PLAN  SUPPLEMENT 

PERSPECTIVE  VIEWS 
AND  FLOOR  PLANS 

of   Fifty    Low    and 
Medium  Priced  Houses 


FULL    AND    COMPLBTB    WORKING    PLANS    AND    SPECIFICATIONS    OF 

ANV    OF     THESE     HUUSES     WILL     BE     MAILED     AT     THB 

LOW   PRICES    NAMED,  ON    THE    SAME   DAY 

THE  ORDER    IS  RECEIVED. 


Other  Plans 

WH  ILLUSTRATE  IN  ALL  BOOKS  UNDER  THE  AUTHORSHIP  OF  FRED  T. 
HODGSON    FROM   2%    TO    50  PLANS,  NONE   OF  WHICH  ARE 

DUPLICATES  OF  THOSB  ILLUSTRATED  HEREIN. 
FOR   FURTHER  INFORMATION,  ADDRESS  THE  PUBLISHERS. 


SEND    ALL    ORDERS    FOR     PLANS   TO 

FREDERICK  J.    DRAKE   &    COMPANY 

ARCHITFCTURAL   DEPARTMENT 
CHICAGO.  ILL. 


i 


Fifty  House  Designs 


E-l= 


ff 


i>l: 


WITHOUT  EXTRA  COST  to  our 
readers  we  have  added  to  this  and  each 
of  Fred  T.  Hodgson's  books  published 
by  us  the  perspective  view  and  floor 
plans  of  fifty  low  and  medium  priced  houses, 
none  of  which  are  duplicates,  such  as  are  being 
built  by  90  per  cent  of  the  home  builders  of 
to-day.  We  have  given  the  sizes  of  the  houses, 
the  cos*  of  the  plans  and  the  estimated  cost  of 
the  buildings  based  on  favorable  conditions  and 
exclusive  of  plumbing  and  heating. 

The  extremely  low  prices  at  whirh  we  will 
sell  these  complete  working  plans  and  specifi- 
cations make  it  possible  for  everyone  to  have 
a  set  to  be  used,  not  only  as  a  guide  when  build- 
ing, but  also  as  a  convenience  in  getting  bids 
on  the  various  kinds  of  work.  They  can  be 
made  the  basis  of  contract  between  the  con- 
tractor and  the  home  builder.  They  will  save 
mistakes  which  cost  money,  and  they  will  pre- 
vent disputes  which  are  never  settled  satisfac- 
torily to  both  parties.  They  will  save  money 
for  the  contractor,  because  then  it  will  not  be 
necessary  for  the  workman  to  lose  time  waiting 
for  instructions.  We  are  able  to  furnish  these 
complete  plans  at  these  prices  because  we  sell 
sc  many  and  they  are  now  used  in  every  known 
country  of  the  world  where  frame  houses  are 
built. 

The  regular  price  of  these  plans,  when 
ordered  in  the  usual  manner,  is  from  $50.00 
to  $75.00  per  set,  while  our  charge  is  but 
$5.00,  at  the  same  time  furnishing  them 
to  you  more  complete  and  better  bound. 


ALL  OF  OUR  PLANS  are  accurately 
drawn  one-quarter  inch  scale  to  the 
foot. 

We  use  only  the  best  quality  heavy 
Gallia  Blue  Print  Paper  No.  loooX,  taking  every 
precaution  to  have  all  the  blue  prints  of  even 
color  and  every  line  and  figure  perfect  and 
distinct. 

We  furnish  for  a  complete  set  of  plans  : 


::-% 

^ 


FRONT  ELEVATION 

REAR  ELEVATION 

LEFT  ELEVATION 

RIGHT  ELEVATION 

ALL  FLOOR  PLANS 

CELLAR  AND  FOUNDATION  PLANS 

ALL  NECESSARY  INTERIOR  DETAILS 

Specifications  cc  isist  of  several  pages  of 
typewritten  matter,  giving  full  instructions  for 
carrying  out  the  work. 

We  guarantee  all  plans  and  specifications 
to  be  full,  complete  and  accurate  in  every  par- 
ticular. Every  plan  being  designed  and  drawn 
by  a  licensed  architect. 

Our  equipment  is  so  complete  that  we  can 
mail  to  you  the  same  day  the  order  is  received, 
a  complete  set  of  plans  and  specifications  of 
any  house  illustrated  herein. 

Our  large  sales  of  these  plans  demonstrates 
to  us  the  wisdom  of  making  these  very  low 
prices. 


•■«■ 


ADDRFSS   ALL  ORDERS  TO 


FREDERICK  J.  DRAKE  &  CO. 

Arihitectural  Vefiirtment 
CHICAGO.  ILL. 


u 


Ml 

mn 

« 

^^ak 

- » 

t  i 

Ucz.f 

'  i 

'! 

|Bj 

'J. 

|;-SH 

R 

'1^ 

1 

1 

A 


s 


J3 
o 

'2 


¥: 

© 

1 

v^.  .• 

CO 

^Jkt'' 

• 

<-* 

''^/' 

c 

0 

fl-' 

/^ 

M 

«?'•■• 

•3 

i^''' 

c 

0 

^n 

.bf 

^s. 

"55 

o 

§ 

^Z' 

Q 

» 

-2  ^ 

c   i 

is 


;c 


i£ 


»    eS 


to     S    M 


S  I  g 


•r   ^*   §   o 

is 


iiil 


^  • -■= 


ii 


ns.      a 


N^ 


U 

r^ 

0 

•M 

(A 

•M 

** 

*m 

M 

•1 

le 

■  "" 

^^ 

"* 

<M 

1 

O 

CC 

V 

J 

• 

0 

^ 

Z, 

« 

c 

u 

u 

? 

<A 

1/ 

^ 

Q 

i"^ 

^■■^ 

S4 

M> 

0^ 

^^ 

>«' 

7f 

1 

«4 

PQ 


ic 

c 

s 

2 


'■J 


LS, 


8| 

.5  -S 

(Q     to 

N     ^  - 


o 

o 

c 

c8 


C 
C 


1^  8. 

o  k.  a 
a  s  u 
®  ^  - 

.2  o.  >. 

'ifl  1^    t* 

8  £5 

a    e«    0 

«•  8  -2    r: 

5  -  <"  -B 
"  J  o    1 


I 
I 


-;•■ 

I  I 

'  s 
I  i 


■^'P 

m 

mM 

1 

s 


u 

'Ja 

s 


r^  2 

CO    c: 
o 

d   V 

P     ^ 

0 

3    J?. 


o 

CO 


,i| 


^^^^^^w7 


TTTsa 


•I 


,  j^ 

I^^B 


i 


S 


H 


CO      «    M 


S. 


js  a 

a  o 


.a  "B 


£.2 


01  ^  a 

■«  *'  .2 

S  3  o 

M  .2  « 

.5  »■:  S 

a  5  " 

Si  o.  H 


8-H 


95 


ijy 


n 

1 

' 

i 

H 

II 

J-.'    " 


i ' 


CO     ~ 

.6  t 


t/3      = 


ffi      t 


3 
j3 


a   S 


.-"    3 


u    a 
(0  -5 


00 


J 


■-   s 


i5  2 

c  o 

I.  " 

c  s 


ca> 


<    -3 

-J    a 
a    o 


o  .S 


in  — 
or  pq 


V 


1 

nsBfi 

■ 

1 

I 

I 


Oi  g 

CO  o 

O  » 

CO  5 


CO 
.  u 

1/3        o 

3      § 

►-H        a 

o 


9 
O 


«flP!m^5^'^!»i»ft^'WV'*!?'»i0«»''5ri.«  J.7  •vr.'.«s«:a; 


OS 

o 

o 

CO 


'^ 


9m 

% 


w^,m^!iSfBmPi» 


s 

US 


C'^ 

"3 

t^ 

i 

o 

— 

CO 

J 

• 

o 

i 

;?; 

^ 

■3 

c 

0 

.bjD 

g 

S3 

*55 

^ 

M 

Q 

« 

i^ 

^ 

^ 

^ 

o 

^r 

t 

C 

n 

3 

L^ 

PQ 

/ 

s 

"ra 

er 

B 

S 

.a 

■1 

is 

** 

•M 

O 

«« 

5 

u 

figMm,^^ii^sx^SL 


^•^A^ 


W.   iii 


I 


«       -2 

©       ■= 
CO         o 


Z^        s 


X 


r^       y 


a     5 


f 


■t^mp:-  :^^:'iMiir'.  .jPiWL^^^eraKrriR^ 


-  m 


a 

'■i 

ao 

f^ 

j^ 

^ 

u 

? 

d 

;?; 

.£JS 


N     J     - 

-  as 


i  c  ••-■  fl " 

«  J5  5  c   S. 

'=  S  ""  2  £ 

e-.2  —  *  5 

4,    e«    5  5  'C 

.2  S  ■■'  >••§ 


i  iFlU'jr  FL0O«.'  PLAN 


5E.C0MD  rLOO;?  PLAH-'jflTX 


i'-mii 


©      - 


O       2 


u  i 


A    ^ 
& 


2 


en 

«§ii;^ 

C   i-i 

«  S 

■^    5g      -     0.     U 

.2  ^    I.  -  --C 

93    CO 

3^   S 

•     ao           s     ft  .~ 

^  - 

0   c  ^  a  " 
00    ft  2    g    a 

2   ® 

II 

•C    =    ft  c    g 

b 

>   ^ 

ft.S  _    «   ^ 

0)    *   5    "    C 
a  "V    a    s    :^     . 

S   5  ^  -2  £  S 

1^    kH 

StCOrtO  FLOOR  PtflN-Jofl, 


f 


^:^jL'i'(PS\.^e.4i:^.rjmk^.x^iii.i'iemh 


k^t^ 


^^V 


e 
o 

2 

«■> 


e 


O 

o 


n 

"a 


In 


-1 


% 


(so 


u 


.  ~^m!tf.%  :x  ^dpi  "SR-^*.*  ' 


■ —.  ^fii     '■■■»■'■■■  .ifc.  »—_  .iJ  ;.  ~  ~*-y  «»  r*i  5-  - '     -t  '     — ■-  «  ■■  ■  ji^- 


■^  '"  '?  . 


.» 


e 
o 

o 

X 

V 


en 

C 
rt 


u 

O 

o 


1 


^ 

flooi 
plan 

**• 

c         o 

i«        •• 

a        S 

c        -p 

1         § 

d  founda 
ns  with  ( 

sS^. 

Si 
Width, 
Length 

cellar  an 

on. 

pecificatlo 

v.=      n 

St   0 

leva 
tten 

M  «     r 

s  con 
side 

typew 

srint 
and 

lete 

rr~    o> 

Sg     E 

CqA     tS 

f^ 


^ai.,^-^^'^ 


s 

a.  ^ 


zn 


rrr'JS/ 


Ms 

P 

0.(0 


8 


o 

O 


.•trt4,7««, 


be 


O 


55   -5  "S. 


M 


•a  ^ 


a  - 


Ri    rt 


03  O 


L 


j 

m 

If 

-  i 

BrP' 

H[fii' ' 

■fl''''' 

V  ' 

i 

\ 

Ll.  ._._..         ' 

a.  s 


b] 


.^£^S 


i#.  *^ 


2  -^ 
b4  r 


UJ 
N 

10    xf 


2    c 


14  o  c      e 
!2—       r  ° 


C    C    o 


^  « 


-  c  c  5  -  o 

-  -2  iS  0  «  2 
p.c5  o.„5^ 

o  c  o  «*  q  a 

3    3  ^    C    C    * 

iv?  o  t:  "D  o  (0 

00  "i-  ™  K  O  c 

c 

TJ-OTJ        — 

CSC  (D 

n  (1  ttf       o 


I*  J 


MICROCOPY   RESOLUTION   TEST   CHART 

(ANSI  and  ISO  TEST  CHART  No.  2) 


1.0 


I.I 


1.25 


■  50 

i^ 

IM 

Li 

1^ 
1^ 

IP 

III  1.8 

J  /APPLIED  IIVMGE     Inc 

^^-  t653   tost    Main   Street 

S*—  Rochester.    Ne«   ''<>"'        '4639       uSA 

.^B  (7t6)   482  -  0300  -  Ptiore 

aas  (716)   288  -  5989  -  Fa. 


te^ll|iiwMt^^t_ 


i^^^m:  ::^T<.7M'^^ms^^^^"'rm- 


•o 

c 


Q 


CO 

c 


O 

o 


«  s 

iqs 

ro 

j=  j= 

T3      M 

»J 

c 
p 


c 
(« 


o 

o 


« 

c      . 

■^§ 

o  >~ 
C     O 

c    •J 

rt  « 

"5.  -c 
c  rt 
o    «> 

■^  •£ 
•a  -r 
c    f 

3     M 


™    o 

V    a, 
o    n 


4)     p 

03  O 


•a 

:i 


V 

o 

5 


CO 

C 

o 
o 

o.  - 

W  CS 

•tJ  ** 

rt  3 

!a^ 

rt  « 

a.  ^ 

<u  v 

4)  3 

O.  O 

E  -^ 

o  '^ 

•a  ^ 

rt  ° 

=  i/l 

D  O 

a.  J 


■J^^^%:.- 


:'irrM*f'\  ^« 


.-•.it^:>^'  i: 


o 


O     (4 

5    « 


O      01 


C4     (« 


o  c 

■^  *> 

n  Zi 

'n  C 

<=  > 

o  V 

o  S. 


OQ  U 


T?3gBl!3 


-,Ji.'; 


HI 


liv 


V    ^ 


2    c 


=    o 


rt   .2 


V 


CD  O 


:  i   )l 


.*?«a'^- 


■jtfi 


o 

U 


en 


U 
N 

55   ■£ 


«    S  o 

^  ^  a. 

M  ti  "~ 

CO  '*■  o 

•   J-  o 

jz  £  > 

u 


1^ 


B 
O 


n 

c     . 
ft    n 

•5.S 


•3.-g 

g  2 

c  * 

3  m 

°  c 

*"  o 

-a  s 

c  rt 

rt  o 

(4  O 

~  u 

V  a 

U  n 

o  u 


8  a 


c    « 


—    o 
CQ  O 


II 


-.jiwS».^'#KI»IB™F«'?*^T5^??aS3r?:3aPi^ 


^11 

1 

mm 

I 

"fMt 


-a 
o 
o 


V 


4)     4) 

a  2 


o 
o 

c 


P    ° 
U.  O 


^■^-j^rja 


I 


ft 


^''^^Ml^i% 


Mt^i^f^  :^'^T^'m^^^ 


u 

9 


W) 


u 

N  -     r- 

2   c 


o 

I 

Ob         c 
5  "3.      5 


gi 

§  '^ 

c    " 
o 


*^    O  Q. 


..   _        E   « 
—         cow 


w  s 


OJ 


2  O 


tl! 


»q 


ir 


■niaiiimKiXi 


h 


V^-^^'j;^.'  1^  h  ff^  \  i 


■Ffis;::':^«!!3i^^^KiJi^r 


•*j^'%.' 


S  ^^If-'. 


■^!^'^a^i 


V 


a.r- 


LI 


s^?""^'^'  n^-. 


kj 


mtii 


Hijii 

\Mm 

^^^^^^^^Tnoi 

mm 

i.^ 


'.t'^';rw.^9^^- 


s 


o 
o 


Ctf 

B 


o 


C/l 

C 


o 
o 


UJ  "Q  ^ 


T3 


>    .J 


ol     i 

J= 

■g    « 

'? 

5  « 

c 

Tl     " 

o 

S    "^ 

(4 

c«    n 

O 

u    nl 

Si 

s 

K- 

o    .. 

~    c 

M      I« 

.tJ 

c'    "• 

> 

8-S 

«    o 

c 

u    cu 

^s 

4J     *^ 

n    '' 

in 

F   « 

t: 

o     " 

w  , 

o 

U  j: 

o 

nl 

> 

1 


> 


i 

1 

i 

1 

:.  . 

i; 

'  j 

if 

i 

U    C 


o 

Full  and  complete 
plans  and  specifi- 
cations of  this 
house  will  be  fur- 
nished for  $6.00. 

Cost  of  this  house 
is  from  $3,650  io 
$3,700. 

■»«!»I*Ymi-,  .". 


C 


C0 

-c 


o 
o 


tu  o  g 
?  -J 


•i 

c 
o 

> 

JO 

"S 
t> 

•o 
w 

•a 

c 
ft 

c 
o 


c 
"5. 

o 
o 


3  s 

^   o 


's.-g 
g  S 
■^:§ 

C  !* 
3     W 

£  1= 

"-  O 

T3  — 

C  <« 

CD  O 

>-  S 
cfl     y 

:=:  u 
(u  o. 
u    n 

o    u 


c    > 

n   i> 

c    »> 

i.'    I 

03  O 


T  III  lYIMllTf  r'  TTli  lilHIll 


AJi 


.3 

o 

B 

O 
u 


i-tOmmahMi 


a 
o 

4-1 
CO 

o 

U 


c 


o 

o 


J! 

S    8  5^ 

35   4  4 

5j 


J) 

o   9 

^  8 
p  j- 


c    • 

JS  JS 

•^■^ 

22 
.     o 

c  '5 

O     rt 

c    > 
"   & 


•2   9 
10  o 


U 


ir«-Kr>    i-Tmrir*:; 


c 

."3 
« 

O 


\v 


Mia  f-=^^^'^u-:'* 


•-*.   ,M)fl^^      -■ 


«  o 
S  2 


U^»^T    \S"'~«.f   --'A-VX^     Q.S 


■3.0 

E  -^ 

o  J2 

•a  r 

S  ° 

^^  ■«-• 

"3  o 


^.m^^ 


u 


S  8 

•S     M 


^ 
£ 


i    » 


o   a 


c« 


•2  c 
o 

s  « 

(4    o 


JS    u 


?    "s   c 


"5  o 


m  o 


i'  ^-'F-  ;r^v%^^?Hpp^i 


:%iMf-' 


is 
i 


8 


v\Nx 


k SViJr"^ ;'  I    §'■''     J2  o 


.4 


CA 
C/l 

3 


u 


M 


■St  "3 

4)  3 

"S.  2 

O  M 

"  :c 

•o  ~' 

rt  ° 

S  to 

3  o 

tfc  O 


;«^^  iv^.55^p*i^B^^ri^^ 


I 

9 
V 

(2 


d 


u 
O 

o 


Cti 


!l 


o  a 

■2  •» 

••'     o 

c    o 

M     ul 

Q.  JS 

_     O 

C     (4 

o    o 

^:§ 

c    > 

Is 

•a  -^ 

c    •« 

M     u 

«     O 

•s  £ 

z 

3 

a. 

ly. 

i:    » 

o 

o.  s 

s 

S   E 

u. 

m   rl 

immmi. 


(94 


u 


N 


s  = 


to    t^ 


a  ^ 


s 


C     •? 


c  o 


IMBgagfc'-'Sf  iMuJKbII' I «"  ~^tVI 


P^^HP'^afET 


1 

n 
o -H 

So 
1.  V) 


■■■-■■'. 
*.(', 


c 
o 

c 

> 

V 

JS 

h 


3 
JQ 


I 


C 
O 

c 

> 


^J 

(U 

vD    ^ 

N 

tN    ^ 

VI 

J=    ^ 

s  ^ 

>J5 

o 
o 


i 


- 

^  ?  .3  9 
a:  £  '^  ''^ 

^    0   X               ^ 

■  J 


o 


-C    » 


c    ■=  - 
'3     u>    tt 

7  g.s 


c    ao 


.S  E 


ii  3' 

Q.    O 


.-a   o 


U.  U 


o 

o 

V) 
U) 
Q 

U. 
O 

en 
< 
flu 

O 

o 
-1 
bu 


« 

u 


^  c,  a 


c 


c 

b. 

J2 

u 

a 

re 

u 

E 

<C 

c 
a; 

«-• 

c 

*^ 

^ 

u 

a. 

c 

a 

4^ 

o 

u 

W. 

i» 

0 

U3 

m 

•o 

11 

c 

tc 

rt 

re 
a 

«rf 

c 

«   « 

c   iJ 

,*"' 

o  -o 

« 

^    u 

4^ 

n 
o 

3    w 

re 

O     C 

«*-■   ••-• 

0     k. 

0 

*-• 

J 

re 

'X 

VI      tfl 

'•7 

c 

0 

u 

3 

o 

1-4 

m  — 

c 

—    re 

c 

re 
u 

2 

'^  s 

^ 

re 

M 

ij    _ 

o 

3    m 

u 

—    c 

Q. 

PQ    _0 

yj 

»JU« 


!  6  •,■'■■ 


W  1 


U  '"•5 


Pi  « 


CO 

o 
o 


o 
z 


^j= 


■*     c 


X 


t_     « 


u  ?! 


•c  — 
c  <» 

re  ^ 


tL" 


-^    O 


C-  2 


re  c 


U,U 


10 


a: 

o 
o 


1 

-1 

u 

J 

^ 

« 

Q 

Ji 

z 

. 

tar 

o 
J 

c 

U4 

k. 

tn 

«N-t 

73 

2 
O 

0) 

Q 

(JU 

O 

CO 

< 

o 
o 

u. 


00 


o 
2 


N 


u 

3;  >i    ~ 


-S    ^  ^ 
—    r    u 


-c        S 

C            •— 

oj        E 

Ul 

■?    s 

c          «< 

«       .t; 

1^ 

t^        > 

>. 

C    j/    - 

"H.  2  '° 

""  «j  »i 

C  -U     U 

0    ^    bt 

•—      Ui      -. 

S.2  Ji 

"O    ^ 

c    I'    >- 

=  H   c 

^    •-     01 

>.  'i 

•a  >-  w 

C     fl 

«  S?   = 

I-   H   0 

ra    "  x: 

u  —  ^ 

^             0              (A 

ai  r 

-i      t^  ^  ■? 

U-  X 

=-     •«  ™  0 

S    m    u 

0    c 

!?=T1 

"  1   c 

1 

X    ^    0 
«    0  •- 

0 

C     i,     *-• 

""'  —  S 

0 

u    ;?  'o 

f 

■■■'-      :l    i    di 

J 

=1£ 

c 

0 

^HR 

T 

fi 


o 


19 


& 


^    u 


—   a 


is   w.  o 


C      -   2 


c-C-: 


be 


CO 


^    o 


S  f 


(«    c 


U,  U 


o 


o 

(0 

< 

r< 
o 
o 


^     j:    ■£ 
\2    c 


CJ 

a. 

tr. 

*-* 

•c 

.-« 

CO 

u 

k. 

t/l 

'— 

re 

»-«i 

r/ 

>. 

«' 

~ 

c 

^ 

i< 

c 

U 

is 

o 

■u 

4-* 

re 

13 

rs 

Q 

c 

c* 

x 

5^ 

3 

c 

fB 

_J 

0 

^. 

im 

(/J 

X 

j_, 

m 

o 

X 

n 

o 

>4 

If) 

c 

'■J 

0 

u. 

u 

c 

v; 

H 

— 

c 

:/) 

T 

rK 

c 

4>J 

ti. 

c 

c 

u 

p: 

-. 

« 

1/3 

i; 

K 

c 
o 

o 

Si 

iT 

o 


o 


S 


f       so 

$     ■=  -2 

-•     .a  J3 


i«    bii 


S  <» 


w,   «> 


ti<  u 


21 


g 


o  — 
—  -g 


ja  .S 


3    nj 


c  5£ 

Z  ^   o 

-    c    c 


-c    U 


5   C.S 


a2 


J*    o 


c.  c 


9   w 


T^ 


M 

o' 

Z 

Z 

o 

•>« 
CO 

Ui 
Q 

U. 
O 

to 

z 

< 

cu 

o 
o 


.J 

a. 


•J 

lb 


u 

• 

^ 

k. 

X 

li^ 

Ij 

^ 

•«> 

n; 

^1 

w 

^ 

-• 

O 

^ 

^ 

> 

^ 

•^ 

t/) 

7B 

c 

"u 

:^ 

-J 

n 

/ 

*• 

-pp 

^ 

.-1 

a, 

as 
O 
O 
-< 

b 

H 


It 
> 

u 

'w 

o 


a 
o 


O  in 

U  C 

7;  19 


"2 

§     i 


..    .   Q- 


be 


5   c 


c 

—  K 

c    n:  ^ 

r:    tf]  M 

tfj  — 

L.    tj  ;: 

re 


r: 


u        ^ 


o 


"  »- 


in    ^ 
o 

-.  ^ 

*  ?^ 

a. 

s 
o 


It 


8 


0     3 


•9  - 


^ 


5  i  " 


w  '-E   'S^ 


o    c 


£  8 

^-    r- 

•2  i 


CS    c 


fii  U 


25 


o 


o 
2 

Z 

o 

mm 
(/) 

(d 
Q 

U. 
O 

CA 

Z 
< 

§ 


X 

1- 

s 

V 

/. 

. 

- 

^ 

-ti 

tt 

- 

r,     -     ■/. 

£     «     C 


T-   c  ii   >. 


=  -J  v; 

S    '    -    c 


2 


5=     =  ? 


~    c    c  — 


^   5!:^    ?    - 


—  r;  f.   '■J. 


VjWp^:^nt?pm' 


'f\ 


■«.■• 

t: 

-l 

w 

C 

u 

« 

2 

o 

en 

« 

• 

>-' 

c 

y 

IC 

fii  a. 

<N 

ll 

c 

vT 

o 
o 

d 


1 

^t 

1 

1 

=   c 

1 

1 

■'    >. 

ij    .ti 

Cfl     ' 

.    -     a! 

,    t-    -    u 

/  i  ;  - 

.2    w 

_•  ji:  j: 

*•-•*-* 

X  ^    - 

.     =   ■" 

y. 

-   c   c 

K  -B  ^ 

,y  s 

w   -    <-i 

«   o   t' 

:^   u     . 

c   ^o 

I                  1- 

■0   ^ 

S   CO 

7]      *^ 

c  o 

-  o 

"    1.-^ 

u* 

c   s 

/        -^    I 

1         >-    ■- 

1          -   ■« 

/            .— 

r     ?  •/! 

•  .M 

S    :j 

Cv     i/i 

c  -^ 

C        £/) 

y     .- 

-c  *^ 

c  ^ 

(4     0 

—       4-1 

— '        t/J 

3     O 

fc  u 


27 


sr 


'.m 


^.:m. 


r 


"T-l 


Mi  i=  I  ^ 


2(5 


..*_i;«i. 


o 

o 
as 


y. 


S 


3 

•c  .2 

CO 

■a  s 
—    c 

*"^ 

(/)  — 

3    ra 


to     t£ 

Co 

o  — 


18 

^    0 


E  - 

C  T 

(J  .- 

C  >~ 


t^ 


^•mim 


B 


o 


c 
2 

a 

>-i 

to 

U] 
Q 

b 
O 

< 
cu 

o 
o 


ir. 


o 


« 
s 

u 


'* 

01 

e 
.2 

> 
u 

II 

II 


5         " 


»_        o 


c 
o 

o    k-  w 

O  C 

j2  "C  o 

•c  c  3 

C  •"  ij; 

41  £r'o 

=    r:  4; 

CO     !/.  ,; 

e 


>-   ■!»:. 


VA_ 


urn 


'*  *V1 


Remember 

We  can  mail  out  the  same  day  we  receive  the  ordei 
any  complete  set  of  working  plans  and  specifications 
we  illustrate  in  this  book. 

Remember  also 

That,  if  you  are  going  to  build,  complete  working 
•^lans  and  specifications  always 

Save  Money 

for  both  the  owner  and  contractor. 

They  prevent  mistakes  and  disputes. 

They  save  time  and  money- 

They  tell   you   what  you  will   get   a.id  what   yon 
ire  to  do. 


m 


-mii 


Estimated   Cost 


It  is  impossible  for  any  one  to  estimate  the  cost  of  a 
building  and  have  the  figures  hold  good  in  all  sections 
of  the  country. 

We  do  not  claim  to  be  able  to  do  it. 

The  estimated  cost  of  the  houses  we  illustrate  is 
based  on  the  most  favorable  conditions  in  all  respects 
and  does  not  include  Plumbing  and  Heating, 

Possibly  these  houses  ^annot  be  built  hy  you  at  tht 
prices  we  name  because  we  have  used  minimum  materia? 
and  labor  prices  as  our  basis. 

The  home  builder  should  consult  the  Lumber 
Dealer,  the  Hardware  Dealer,  and  the  Reliable  Con- 
tractors of  his  town.  Their  knowledge  of  conditions 
in  your  particular  locality  makes  them,  and  them  only, 
capable  of  making  you  a  correct  estimate  of  the  cost 


.-..f  % :  "m^.J>Mm4» 


/.% 


TO  CORRECT  MEASUREMENTS  of  areas  and 
cubic  contents  in  all  matters  relating  to  buildings  of  any 
kind.  Illustrated  with  numerous  diagrams,  sketches  and 
examples  showing  how  various  and  intricate  measure- 
ments should  be  tak«n     ::     ::     ::     ::     ::     ::     ::     ::     :: 

By  Fred  T.  Hodgson,  Architect,  and  W.  M.  Brown,  C.E.  and  Quantity  Surveyor 


m 


/jiHIS  is  a  real  prartir.il  bonk, 
^*'  phi'wing  h'W  all  kinds  of 
odd,  (  iiMikfcd  and  ditticiili  mea^^- 
ui  t'liif  nts  in  a  y  Ix;  takc-n  to 
^e^ult•  cnritTt  rc-^nlts.  Tliis 
wnik  in  no  wav  rontiirts  with 
any  wmk  on  t'-^tunalint;  as  it 
dtjfs  nut  nwv  piire-i.  nt'ithfr 
does  il  attempt  ta  deal  with 
MUestinns  of  lal.itr  or  e^tiuiato 
lii'vv  ninrh  tlir  I'xrciuion  of  rcr- 
lainwuKs  will  co-t.  It  sjniply 
dt-aU  with  tliM  <|Mes,iions  nf 
areas  and  riibic  r* intents  of  anv 
1,'iven  wt'ik  and  shows  liow 
liifir  ait-as  .uid  rmittMils  may 
i*-;idily  be  oliiained  and  fin- 
ni-hes  for  ttie  reji'ilar  estimator 
tluj  data  npon  whi^h  he  can 
b.i'-t:  liis  prices.  In  fact,  the 
U'lT  k  is  a  ureat  aitl  and  .issist- 
ant  to  the  re^nilar  estimator 
ami  of  inestimable  vahie  to  the 
g'jueral  builder  and  contractor. 


■* 'S'l 


12mo,  cloth,  300  pages,  fully  illustrated,  price     -     $1.50 

Sold  by  Booksellrrs   ffenerally  or  sent  postpaid  to 
any  address  upon  receipt  of  price  by  the  Publishers 

FREDERICK  J.  DRAKE  &  CO. 


PUBLISHERS 


CHICAGO.  U.S.A. 


*  y\ 


